Naked Science Forum

On the Lighter Side => New Theories => Topic started by: jeffreyH on 17/07/2014 01:17:45

Title: Lambert's Cosine Law
Post by: jeffreyH on 17/07/2014 01:17:45
I am looking to adapt Lambert's Cosine Law for the gravitational field. What do others think? Is this already done? Can it be done?
Title: Re: Lambert's Cosine Law
Post by: JP on 17/07/2014 02:43:12
It has to do with incoherent light coming off a perfectly reflecting surface.  What is the motivation for applying it to gravity?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/07/2014 21:23:45
It has to do with incoherent light coming off a perfectly reflecting surface.  What is the motivation for applying it to gravity?

It is the way you can relate area to particle emissions per second. See the section "Details of equal brightness effect" at wikipedia.

http://en.wikipedia.org/wiki/Lambert's_cosine_law

If gravity has force carrying particles there may be something to be gained by investigating a relationship. While it is nonsense to talk of luminosity for gravity field strength over an area is relevant. Field strength comes down to particle density. I have considered an equilibrium point between electromagnetism and gravitation with increased mass sizes when considering collapsing systems.
Title: Re: Lambert's Cosine Law
Post by: JP on 18/07/2014 04:06:17
...which is fine so long as you're modeling particles that follow Maxwell's equations and add incoherently (they don't interfere with each other). 

But that's a pretty bold assumption unless you're dealing with incoherent light.  And as a bold assumption, it needs strong evidence to support it, so why do so for gravity?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 19/07/2014 19:34:58
...which is fine so long as you're modeling particles that follow Maxwell's equations and add incoherently (they don't interfere with each other). 

But that's a pretty bold assumption unless you're dealing with incoherent light.  And as a bold assumption, it needs strong evidence to support it, so why do so for gravity?

I have absolutely no answer to that. I have thoughts on the subject and I am working on theories but that is speculative. Self-interaction is a big issue here and is a theory killer in many respects.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 12:55:30
If we go back to Maxwell we find c = 1/2b5d72d8579df6acc40aea77a68cac77.gif. This can then be modified to represent C^2 = 1/fefdcd9c3807378ab2ae0a72566a5194.gif or fefdcd9c3807378ab2ae0a72566a5194.gif^-1. For rs we could then produce 2GM/fefdcd9c3807378ab2ae0a72566a5194.gif^-1. This may seem like an insignificant substitution. Excuse the very bad latex. What if we make rs a unit radius by setting it to a value of 1? Then the only unknown is the mass which can be calculated by re-arrangement.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 13:17:07
We may now be able to link this to the curl equations of the electromagnetic field. It may be more productive to set the mass equal to the Planck mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 15:01:54
This then results in the relationship:

(epsilon0*mu0)^-1 = 2*G*Pm/2*Pl

It should follow that:

epsilon0mu0 = Pl/G*Pm

NOTE: To substitute in the curl equations we have basically halved the gravitational component by canceling both factors of two. This then balances the dual equation and could indicate two separate gravitational components to the interaction.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 16:00:10
I haven't really checked any of this and it may be hogwash. If right it may already be known anyway and have no significance. At this point I do not have the information to draw conclusions.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 16:18:09
Now back to Lambert's Cosine Law. This would only be valid if the rates of emission between photons and gravitons differed in a distinct proportion yet to be determined. This of course relates back to the work on black body radiation and has a strict set of boundary conditions.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 17:59:50
We have two curl functions

-c8cf0e702624773dd1dfd0821a2a8bfa.gif
-766743128098f83ed76ebbe43f076753.gif

Which can be substituted with

-e8b40a892f462413b0c8d876775c483b.gif
-096e31371fcc75f544a6e1aa4dfc126c.gif

Here we can consider Pl to be a fixed constant and then the only variable quantity in the first term is the mass represented by Pm. This can not be increase otherwise we violate the Schwarzschild metric. This should then run from some lower bound up to the Planck Mass. What this lower bound should be is also undetermined. This then introduces a density variation into the curl equations. Is this valid? I don't know.

NOTE: This does however tie us to the Planck scale. This is where we need to be to produce any kind of theory of quantum gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/07/2014 23:06:30
The reduction in mass should coincide with a reduction in escape velocity. What effect would a reduction in mass mean for epsilon and mu (see image). Actually this is the wrong way round. This should actually model how light slows down in an intensifying gravitational field. It should model redshift. I need to delve deeper to make sure this is true.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/07/2014 00:56:29
In our mu epsilon equivalence our Planck length is the numerator and appears in the place of time. As light speed is 1 Planck length in one Planck time this appears reasonable. We are also relating our mass to the Coulomb field. The epsilon mu value will increase with decreasing mass and decrease with increasing mass. As was stated previously the mass value can only decrease as increasing this will violate the speed of light limit. We would be calculating using values greater than c which is prohibited.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/07/2014 01:13:38
It may be only the permittivity that is affected by changes in mass. This would explain the reduction in reactivity expressed as time dilation. The substitution in the curl equation is still incorrect as this does not express the limit at an event horizon so more work needs to be done to correct this. What the missing factor is remains unclear.
Title: Re: Lambert's Cosine Law
Post by: JP on 22/07/2014 03:26:27
I've moved this topic as its veered well into New Theory territory.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/07/2014 20:42:45
To revisit this it can be shown that f(z-ct) and f(z+ct) will satify the differential equation. In -e8b40a892f462413b0c8d876775c483b.gif Pl and 18a391e1e6224c9204d7efb548cced66.gif should be at the same scale so to convert this we need to assume t to equal 1 Planck time. Then e858822aef35012745f422909cd47bbf.gif needs to be adjusted accordingly. This then sets our wave equation in the environment of the black hole event horizon. Or at least I think it does. I will be investigating this. Both E and H the electric and magnetic components have three separate parts in x, y and z. This is where the difficulties may arise.

NOTE: Treating this as a scalar wave equation is not very useful. Having the wave expressed in only the z direction and holding x and y at zero just won't describe reality in extreme conditions. While this is how Maxwell linked c to light this should be discarded. The resulting formula is more complex but defining the electromagnetic wave and its path near an event horizon should be a good starting point for a path into quantum gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/07/2014 21:46:54
I have a new method to add to this thread that plots in 3 dimensions with an implicit time function that will describe the progression of a wave with shifting of the waveform and time dilation incorporated. That will have to wait until the weekend. A scalar time function can also be applied to the x, y and z dimensions to show the shifts from blue to red and visa versa. This is then expressed in terms of the gravitational field. This is modeled at the quantum scale using an implicit unit vector for time. It can then be applied macroscopically to planetary motion.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/07/2014 11:06:26
Before trying to link the gravitational field to Lambert's Cosine Law I need to take a detour. This starts with the unit sphere and the unit circle. Using the unit sphere and circle shows some interesting relationships and can be scaled up. This can then be used to describe both subatomic and macroscopic domains.

The circumference of the unit circle is 2*pi. To determine the angle of an arc around the circle whose arc length is equal to the radius we can use (1/2*pi)*360 which can be simplified to 7/44*360. This proportionality will become important when viewing interactions at differing scales and relates to wave frequency, length contraction and time dilation effects. The angle we have determined can be converted to radians to use in calculations.

It is interesting to note that the period of sin x is 2*pi. This can be utilized by considering forward motion and angular rotation as it relates to the unit sphere. The relationship between these two properties can describe the evolution of a wave and can be related directly to the gravitational field. When used it can be shown to show the underlying mechanism of the Pauli Exclusion Principle and the difference in energy levels required between electrons.

There are 3 directions of motion under consideration within this model. One motion is forward direction and is considered to be aligned with the poles of the sphere. The two other directions are angular. The first is around the equator and the second follows a longitudinal path intersecting both poles. The maximum unit of motion in unit time in the polar direction is equal to the unit sphere radius. The maximum unit of motion of the angular paths is 7/44*360 as stated above. If viewed at the Planck scale the angular components cannot reach this speed or none of us would be here. Therefore we can deduce that this dampening in angular momentum must be due to gravity which is what the current physical theories state.

If we follow this line of thinking through to its conclusion we can show that when considering the universe as a whole system light might get infinitesimally near to c but will never actually reach it as long as any gravitational field remains. I will demonstrate the reasons for this conclusion as I proceed.

NOTE: In case this limit on light speed seems absurd consider that if even light cannot reach its own maximum speed then certainly nothing else will. Also all mass is affected in the same proportional way as the photon within an equivalent field and the speed of any one component of the system is relative to everything else. One other consequence of this model may be the discarding of Lorentz transformations and the use of frames of reference. I say may because I have yet to properly determine this.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/07/2014 12:37:36
Before proceeding further it would be useful to consider a thought experiment. When considering planetary motion we encounter the elliptical orbit as found by Kepler and later explained by Newton. From the relationship that equal areas of the orbit based at one focal point are equal for equal time periods we can imagine a scenario with planets. Consider a planet tipped on its side so that both poles align with the path of the orbit. The planet is then spinning around the orbital path as it moves. If we imagine a point on the spinning equator that we follow as it describes a path centered on the orbital path we see that frequency of the wave increases with distance from the focal point. We can then show that we have a redshift described nearer the object and a blue shift described as the planet moves further away. Light does not behave this way as shown by the receding galaxies indicating an expansion of the universe. The model being described here will eventually explain this.

NOTE: This thought experiment disregards the time component of the wave in case anyone was going to object. That was on purpose and will be explained later.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/07/2014 14:33:53
This model will ultimately describe two types of photon. One with diverging electric and magnetic fields and the other with converging electric and magnetic fields. The type with diverging fields will propagate away at c while type with the converging fields has lower velocity, is imbalanced and will converge with a positive field. The converging photon is more severely affected by gravitation and interacts with it differently and may explain levitation in superconductors.

NOTE: The photon with converging fields may be equivalent to the hypothesized magneton. The period of this convergence is pi.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/07/2014 10:23:50
We have a very interesting explanation of how DE Broglie derived his wave equation here:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/De_Broglie_Wavelength

His starting point was to assume E whilst represented by mc^2 could also equal hv which is Planck's constant times velocity. This term operates for light only so examining equation 5 we see that he ended up with essentially h/mv. If we take angular momentum as a stand in for energy and because we have 2 fields in motion we should be able to derive an equation at the Planck scale that equals De Broglie's. This is dependent upon the radius of a particle.

It has to be understood that we are no longer describing the wavelength here. If we are working in natural units the angular momentum will range between 0 and 1. The velocity is forward momentum and so describes the stretch of the wave with time. Hench this formula describes the rate of change in the wavelength.

Now if we go back to e8b40a892f462413b0c8d876775c483b.gif we also have a mass component. If we have Va as our angular momentum we have h/Va*v as the velocity of a wavelength segment which can then be related to the time over which it evolves. As 9c2a198759772ec564bcb0cd72619086.gif describes our wave and 592c6c14f5632724e5b255bf0454a4fc.gif relates this to mass and gravity we should be able to combine these. Remember that the mass was the only true variable in 592c6c14f5632724e5b255bf0454a4fc.gif and in the De Broglie equation it is mass we have modified by replacing it with angular momentum.

Since we are only modifying mass all our calculations are done at a set radials distance as mass dereases. This is because Pl never changes. If we take the earth as an example we need to correlate a change between the surface and twice the radius to our fixed radius. We need to see how the mass should decrease to give us the right gravitational field strength. This factor is important to determine. Because we hold the radius fixed we can then model the wave evolution over a spherical surface as described above. This is not meant to reflect the reality of particles but is a device to determine the relationships.

In our left hand term the mass factor relates to the radial distance the wave travels away from the source. This has to be a density change. It is not only related to the density of the source mass but also the gravitational field density at any radial distance. Our factor will be p with p= 1/r1^2 where r1 is the distance away from the source hence we now have 502f5f044c5097f62cb4a47075387275.gif as our first term. The use of radial distance now connects the permittivity/permeability term directly to the wave evolution via the vector r1. The starting point will have r0 (our static radius) equal to r1. As r1 moves away the density is reduced by our term p. This will produce a gradual blue shift and can be related to time dilation via an implicit time on the z-axis. This does NOT describe the evolution of a light wave. Other factors come into play for light.

From now on the model uses linear algebra.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2014 17:50:20
At this point I need to state one conclusion I have already reached. The magnetic field force carrier should be a symmetry broken photon. It needs a partner particle which I firmly believe is a symmetry broken graviton. We should be thinking in terms of a gravitomagnetic field as well as an electromagnetic one. Now all I have to do is prove it.  [8D]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2014 20:36:37
The next posts will be concentrating on examining the effects on the partial differentials for E and H in the Maxwell equations.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2014 21:39:59
I have changed the first sentence of paragraph 2 of reply #17 from "The diameter of the unit circle is 2*pi" to "The circumference of the unit circle is 2*pi" as the former was in error and was confusing.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2014 22:15:12
At this point 3 things about the De Broglie equations should be understood.

Frequency and Energy are directly proportional.
Wavelength and Energy are inversely proportional.
Wavelength and frequency are inversely proportional.

Angular momentum was substituted for energy above so for our model evolving a wave over a spherically rotating and forward moving surface we can say:

Frequency and Angular Momentum are directly proportional.
Wavelength and Angular Momentum are inversely proportional.
Wavelength and frequency are inversely proportional.


Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/08/2014 00:46:53
This entry in wikipedia describes the general direction in which I am going.

http://en.wikipedia.org/wiki/Gravitoelectromagnetism
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/08/2014 21:55:53
I think I've found the mass of the photon by finding the point of unity in one of the factors in the functions I am examining. I have no proof that this is correct but I'm posting it here in case it is right and it can be recorded. The value is 2.42169*10^-25. The units are eV/c^2
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/08/2014 21:19:55
I am going to ignore the previous post for now as I am not convinced by my methodology. For consistency with standard definitions measurements in the Planck scale will now be donated by lP for Planck length, tP for Planck time and mP for Planck mass. A correction may need to be made to our factor p. It may be that this should possibly be r0/r1^2. Unity may not be the correct scaling for the denominator. It would appear that the former value would produce too high a gravitational field strength. It is known that gravitation is a much weaker force than electromagnetism.

Our partial differential has an implicit time on the z-axis so I need to investigate the implications for the denominator as it is not really momentum that this would represent as such. This uses time like a one dimensional space component that is unidirectional. When applied individually to the x and y axes this would represent a combination of length contraction and time dilation. The z axis should remain unaffected to preserve a direct connection to the speed of light.

If we assume r0/r1^2 as our p function then when r0 = r1 we could simplify this to 1/r1. This may be invalid. Also angular momentum and the vector of forward motion need to be treated independently. The upper limit on 2554a2bb846cffd697389e5dc8912759.gif of 7/44*360 needs a proportionality factor of its own.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/08/2014 23:52:45
When we substitute angular momentum in place of mass-energy then any deviation from a straight line path that the wave takes is due to the stress on the waveform. So we have angular momentum and stress instead of mass energy and stress. This is easier to deal with in a mechanical way. We can effectively add a stress energy tensor into the model at some stage as an angular momentum analogue. As frequency and energy are directly proportional and substituting angular momentum for energy also gives a directly proportional relationship we can put in place the last link between particles, stress-energy and gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/08/2014 00:10:33
One very spooky conclusion that can be reached is the quantization of momentum itself. This would explain the fact that zero point energy remains at absolute zero and could be the basis for this quantization.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/08/2014 01:47:26
If momentum is quantized then there is a relationship between the kinetic energy of a particles motion through space and the angular momentum of the particle waveform. If slowed due to time dilation then there will come a point where the minimum quanta of angular momentum is reached and can no longer slow down any more. If this equates to the speed of light in the direction of travel then the quantization of momentum is the cosmic brake rather than light speed only. There will be a minimum quanta of time dilation that equates to light speed. This would put an end to speculation on FLT travel.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/08/2014 19:09:56
An interview with Roger Penrose that any serious physicist should read and absorb.

http://discovermagazine.com/2009/sep/06-discover-interview-roger-penrose-says-physics-is-wrong-string-theory-quantum-mechanics
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 07/08/2014 18:39:49
The uncertainty principle has been questioned many times, particularly the Copenhagen interpretation. I find that the uncertainty principle as a concept is quite valid but the number of unknowns involved makes the measurement uncertain. It stems from a seemingly deterministic system. However it is not possible to determine the point in the evolution of the wave equation you will be measuring in advance. It would necessitate knowing the previous states of all other processes that affected the previous states of the wave that led to its current state at point of measurement.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2014 19:06:56
The next step is producing the mathematics for the non-gravitational portion of the wave function. This will necessitate the use of matrices, one for each partial differential initially.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2014 14:43:53
I have attached 3 plots. The first two take no account of gravitation and show the profiles for diverging and converging waves. The profiles differ for each one. The third is a plot that accounts for the waves shift due to gravity. It is important to note that the profile for both under the influence of gravitation are identical and show a curvature in the progression of the angular quanta only when in a gravitational environment. The straight lines connecting one end of the angular progression to the other should be ignored. This marks the transition past 360 degrees on the x axis.

NOTE: No time element is present in the three graphs shown here. Time, if present, would be shown on the y axis as in Minkowski diagrams. However this would not be a simple unit scale and would vary. The function required to plot this scale is linked directly to the equations of time dilation. The plots of curvature would need adjustment to prevent a negative time appearing during wave evolution.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2014 20:47:36
Now back to our time component and the need for this to be incorporated on the y axis. To do this we need to modify e8b40a892f462413b0c8d876775c483b.gif to become 94b6446995be78aa3c87cf13d02ef3c0.gif. We should now have gravitation operating on time and not velocity. Since velocity (momentum) and gravitation can be considered equivalent in particular cases time makes more sense.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2014 21:09:08
It is almost time to examine Lambert's cosine law again.

http://en.wikipedia.org/wiki/Lambert's_cosine_law

How can this be applied, in a modified form, to the gravitational field? The velocity in the case of light is c and in the case of gravitation is also considered to be c. This then fixes the denominator in our modified Maxwell equation. We therefore arrive at 064a0e8d2d117c657660c1de7cbc0ab4.gif. We can now state our equations as de79d85bb6f9d019009d824eecf0327b.gif. Since we have a range of masses for our denominator (photon mass to Planck mass would be the ideal range) We arrive at c9c092f77d9d436c5dc5800bac0593b5.gif. To model any particular particle we substitute its mass for dm. This will model two of the same types of particles ie two electrons interacting. For interaction based on gravitation we must use the Planck mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/08/2014 00:20:13
The attached image is a purposely exaggerated view of the evolution of a waveform over time. We now have a time scale on the y axis.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/08/2014 01:43:37
It should be noted that in the previous plots the speed of light is violated as gravitational damping is NOT yet applied. This will have an effect on the curvature inherent in the angular quantization. It also implies a direct link between light speed and gravitation. Without gravitation this limit on photon velocity would not exist. This conclusion is tentative at best. This could also indicate an intimate relationship between the photon and the graviton.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/08/2014 20:12:01
With 592c6c14f5632724e5b255bf0454a4fc.gif being equal to 0877f871cf7ea9b700a1c9b1c97392cd.gif we can simplify further to 14e556ac118387864e8a34fbb102e360.gif. This results in the attached plot. Plot no 2 shows this with a log scale on the y axis. These plots can not reflect the reality of gravitational interaction without more work on the mathematics. It is a tentative step. Maybe they will be of use, maybe not.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/08/2014 23:51:57
I have now been able to reformulate the equations to remove the gravitational component altogether. This may seem like no big deal but in fact it is. This plot is attached.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 12/08/2014 06:54:06
If we now integrate this with Maxwell's equations we see a looping in the wave as it loops back on itself. Nearer to a source this flattens and exhibits length contraction in the direction of the force of the field. Taking this looping into account we still see the curvature of angular quanta preserved. Uncertainty is due to a variety of factors including the looping nature of the waveform.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/08/2014 13:26:12
The state of the photon wave, evolving symmetrically or with broken symmetry, is dependent upon the Pauli exclusion principle and the exact spin states of electron pairs as the photon is emitted.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/08/2014 13:30:22
The graviton's loop profile is attached. This diverges from the profile of the photon with increasing distance from the source. This indicates a well defined strict density in the gravitational field in order to trap light. The deflection of light may be due to an induced and partial symmetry breaking of the electromagnetic wave due to interaction with the graviton.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/08/2014 13:52:10
This raises the question, if gravitation can bend light then does light bend the gravitational field. If we view this in terms of Newtons third law the answer appears to be yes. Therefore we can ask, can we focus the gravitational force using light?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/08/2014 12:50:49
I now need to go back and add a gamma factor at the Planck scale where v = 0 to represent the conditions at the event horizon of a black hole. This may take me a while.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/08/2014 14:08:43
Attached is an image of the curvature inherent in the graviton with more realistic frequency values.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/08/2014 14:29:54
I am now going down a slightly different path. In another post JP brought up coherent quantum states which I will now be following. The work here shows a deterministic view of the wave equation which is an approximation to reality. The starting point was artificial and considered frequencies that would not exist in the real world. This does give some insights which I will now try to follow.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/08/2014 00:56:16
It may be that the process of photon absorption may be due to wave inter-locking. When both interacting waves are in the right phase this process can happen. When the phases are not aligned we have reflection or refraction. For materials like glass the structure of the medium is important and generally causes a refraction with a precise angle. In wave interlocking the photon would not so much be absorbed as add momentum to the electron and deflect its path to a higher orbital until re-emitted when the waves disengage. This then gets round the problem of Planck length/Planck time and the photon/electron momentum problem at the Planck scale.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/08/2014 23:16:29
Wave interlocking can be thought of like the table cloth trick where crockery remains in place if the cloth is removed fast enough. However if this is the actual mechanism of photon absorption then it makes describing the interaction of the graviton much harder to derive. The waveform of the gravitational field must be very strange indeed.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 24/08/2014 16:13:51
The first derivative of g = GM/r^2 is 2GM/r^3 which describes the change in acceleration or jerk. The Schwarzschild metric has 2GM/c^2 and the right hand factor of our wave function has 14e556ac118387864e8a34fbb102e360.gif. The right hand term can be rewritten as e5bfed90d20d5182b5047487241e4b3a.gif where r is distance of the wave from the source of gravitation. Before getting back to Lambert's Cosine Law it will be necessary to investigate Coulomb's Law with respect to the previous equations. This may seem like a very strange path to take, which it in fact is, but I want to determine certain relationships before proceeding.

A correction to the above. e5bfed90d20d5182b5047487241e4b3a.gif should be 957559fc6242b50648307d3958ea52c7.gif.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/08/2014 20:08:31
Light is affected by gravity which experiment shows is true. A photon leaving a gravitational field is slowed. Where then does the energy come from to restore the loss of kinetic energy whilst leaving the gravitational field? The photon will gain momentum due to the fall off of gravitational effect due to the inverse square nature. Can it be only the fall in gravitational strength that is the cause? Or is it the relationship between forward and angular momentum which is redistributed according to a law of nature? We say this is the conservation of energy. Then energy IS momentum. Conservation of energy thus means conservation of momentum which again suggests quantization of momentum.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/08/2014 23:15:17
Attached is a graph of the gravitomagnetic wave of the graviton. Both positive and negative energies are presented. Maxwell's concern about negative energies can be overcome if momentum and energy are considered equivalent. The positive and negative energies are then due to opposing spin in a two component particle. The positive and negative elements cause attraction with both positively and negatively charged particles. This double opposing spin also indicates that the graviton cannot be polarised, unlike the photon. This plot is slightly reminiscent of the amplituhedron. I say slightly as it cannot be said that they are equivalent in any way.

The positive portion of the wave exhibits the loop back inherent in the magnetic field. The positive portion has an altogether different profile.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/08/2014 02:07:30
I have come to a conclusion driven by the mathematics that I didn't expect. I had assumed that for any black hole the point at which escape velocity reaches c would be at a greater radius than the point where g equals c. This was not the case. Perpendicular motion, if my math is correct, is overcome before the event horizon is reached. I do need to check this but if correct then any photon is doomed before it actually reaches the event horizon. I am still skeptical about this result as it means black holes should consume mass more aggressively than expected. This is not what happened to the G2 gas cloud.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/09/2014 14:56:10
One point to note for anyone even remotely interested in reading this thread. I am breaking mathematical notation in ways that physicists would consider invalid. Don't try this at home because you can't determine why.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/09/2014 22:49:30
The reason why there may be a region at a point outside the event horizon that seals a particles fate may be because magnetic field lines first fall beyond the event horizon making it a sink for the field. To maintain the circulation of the field the particle would be doomed to enter the horizon once the field line is trapped.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/09/2014 00:54:45
For the first time I have been able to plot the profile of acceleration toward light speed taking gravitational self interaction into account. The plot is attached and shows the view from a frame external to the acceleration. This SOLVES the infinite mass problem due to the time dilation effects. The external observer would initially see the object accelerate and then appear to slow down and become fainter. This is similar to the effects when approaching an event horizon.

NOTE: This profile suggests an intensifying gravity well around the accelerating object.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/09/2014 01:17:34
This could mean that during the early stages of the universe some black holes were formed during the inflationary period when velocities were much higher and not simply due to collapse. These would be the so called primordial black holes. This would also explain the apparent slow down after the inflationary period. The other strange thing when viewing the plot is that it suggests that things appear to be accelerating away precisely because we are slowing down and the time dilation and length contraction are now reversing.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/09/2014 02:09:12
Counter to what I stated previously the universe has not actually left its inflationary phase. This is a consequence of gravitation. The gravitational field evolves away from the source into an expanding spacetime. The energy of the field occupies a compressed space near to the source. Gravitation therefore cannot itself be affected by dilation or contraction. The consequence has to be that its propagation is superluminal until it reaches infinity where it will equal the speed of the photon. A consequence of this is that the gravitational field can in fact radiate out of a black hole.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/09/2014 02:13:49
Another consequence of this is that it must be mass itself that compresses the spacetime and not gravitation. The operation of gravity merely accumulates the mass in the first place. I can currently think of no way that gravitational force can be carried by a boson with such properties. It may be a consequence of the spin 2 nature but how I don't know.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/09/2014 02:33:21
The only way a force carrier can appear to operate under these circumstances is by losing mass with distance from source. This is puzzling. However it does suggest that gravitational waves would only be detectable near to a strong source. This also implies that at infinity the particles ceases to exist. If there is a way of focusing gravitation using light then this could be a mechanism for detection.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/09/2014 02:48:34
The loss of mass of the graviton could be the source of dark matter as more mass would be lost nearer the source. In galaxies with central massive black holes this loss of mass would explain the galactic halos. The loss to dark matter/energy at source would also prevent orbital decay due to massive bosons near to the source. This loss would allow the boson to maintain a constant speed free of time dilation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/09/2014 02:53:01
Thinking about the G2 gas cloud encounter brings up another possibility. When a star has lost enough gravitational energy and the amount of dark matter around the object reaches a critical point then collapse could be initiated by pressure from this excess of dark matter. Dense object could then radiate a lower strength gravitational field than currently thought. Since G2 did not exhibit the expected behavior this could be a feasible explanation. The pressure from dark matter/energy would aid in keeping the cloud intact.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/09/2014 19:33:08
The graph of acceleration to light speed can also be applied to black holes. As the time tends to infinity the motion through space tends toward zero. At an infinite distance light will have a velocity of 1 Planck length in 1 Planck time. The knee occurs before half light speed and this point should correspond to a turning point in the tidal forces. These forces then increase exponentially. Calculating the point at which the escape velocity of a black hole equals this value will give the exterior zone of no return. Particles are still free to circulate around the black hole but only light can ever escape this region. This then confines the accretion disk.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/09/2014 20:22:40
When using the graph to calculate the extension out from the event horizon we end up with the equation hc = Rs + 2*pi*Rs. Where hc is the radial distance to the boundary of the confinement zone. Therefore the extension of the radius out to the accretion confinement zone is equal to the circumference of a great circle of the sphere describing the hypothetical perfectly spherical event horizon of a black hole. This is derived directly from the knee of the curve in proportion to the fraction of light speed represented by that point. The fact that this is equivalent to 2 * pi * Rs appears to validate prior assumptions.

NOTE: This hypothesis should be easy to falsify through astronomical observation of dense objects and candidates black holes. This all depends upon being able to reliably determine the extent of the accretion zone.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/09/2014 21:58:25
There is a second point of inflection on the graph nearer to the event horizon which should indicate the start of the ergosphere. This I have yet to calculate. When studying the curve there appears to be no indication of the position of the event horizon itself. This is simply the termination of the curve at infinite time.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/09/2014 22:07:20
The portion of the curve following the start of the ergosphere describes a complex geometry. This indicates that the frame dragging induced in this region is within a complex space. The plot terminates at this point as it involves complex numbers.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/09/2014 00:43:27
There is an error in my method which can be illustrated in the section "2. THE INNER DISK RADIUS" here: http://iopscience.iop.org/0004-637X/476/1/278/fulltext/35151.text.html

The 2*pi*Rs is incorrect. This calculation should instead refer to the spin of the object related to torque. This is due to the scales of distance and time on the graph. The value for confinement radius is therefore incorrect.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/09/2014 23:38:56
The above graph has no torque component at all because it has no relation to angular momentum. There is however some connection to angular momentum in a relativistic manner which at present is undetermined. The axes on the graph should really have been labelled as time dilation over length contraction but I just couldn't be bothered to change them.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/09/2014 01:06:52
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets. As the fuel is exhausted and this mass cools the density increases. This can then cause a vacuum in internal cavities. I have no idea at this point if this would be a valid reason for black hole formation or how it would work. Without a density profile for the interior of a star it is impossible to determine. A combination of internal gravitation within the cavities and the vacuum could be the initial cause of collapse.
Title: Re: Lambert's Cosine Law
Post by: alancalverd on 18/09/2014 06:50:47
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets.
not true. Consider a neutron star

Quote
A typical neutron star has a mass between ~1.4 and about 2 solar masses with a surface temperature of ~6 x 105 Kelvin [3][4][5] (see Chandrasekhar limit).[6][a] Neutron stars have overall densities of 3.7×1017 to 5.9×1017 kg/m3 (2.6×1014 to 4.1×1014 times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of 3×1017 kg/m3.[7] The neutron star's density varies from below 1×109 kg/m3 in the crust - increasing with depth - to above 6×1017 or 8×1017 kg/m3 deeper inside (denser than an atomic nucleus).[8] This density is approximately equivalent to the mass of a Boeing 747 compressed to the size of a small grain of sand. A normal-sized matchbox containing neutron star material would have a mass of approximately 5 billion tonnes or ~1 km³ of Earth rock.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 19/09/2014 18:08:47
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets.
not true. Consider a neutron star

Quote
A typical neutron star has a mass between ~1.4 and about 2 solar masses with a surface temperature of ~6 x 105 Kelvin [3][4][5] (see Chandrasekhar limit).[6][a] Neutron stars have overall densities of 3.7×1017 to 5.9×1017 kg/m3 (2.6×1014 to 4.1×1014 times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of 3×1017 kg/m3.[7] The neutron star's density varies from below 1×109 kg/m3 in the crust - increasing with depth - to above 6×1017 or 8×1017 kg/m3 deeper inside (denser than an atomic nucleus).[8] This density is approximately equivalent to the mass of a Boeing 747 compressed to the size of a small grain of sand. A normal-sized matchbox containing neutron star material would have a mass of approximately 5 billion tonnes or ~1 km³ of Earth rock.

Thanks for the info. It was a bit of cock-eyed speculation really.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/09/2014 00:25:51
Related to the time dilation and length contraction plot is the spacetime density over momentum plot which is attached. As particles reach relativistic velocities the spacetime density increases exponentially.

NOTE: This is linear and not volumetric density. It can be centred at a point in spacetime. The implications of this are that every black hole has an equivalent density due to spacetime density. More massive black holes have the same density due to spacetime contraction as smaller black holes. It is this equivalent density at the event horizon that traps light. This indicates that the event horizon and the singularity are exactly the same thing.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/09/2014 01:23:28
All this leads to the conclusion that recession of galaxies is related to a decreasing spacetime density as a natural consequence of the big bang.
Title: Re: Lambert's Cosine Law
Post by: vampares on 21/09/2014 05:49:36
Trapped in a prism, in a prism of light
 Alone in the darkness, darkness of white
 We fell in love, alone on a stage
 In the reflective age

Entre la nuit, la nuit et l'aurore.
 Entre les royaumes, des vivants et des morts.
 If this is heaven
 I don't know what it's for
 If I can't find you there
 I don't care

I thought, I found a way to enter
 It's just a reflektor (It's just a reflektor)
 I thought, I found the connector
 It's just a reflektor (It's just a reflektor)

Now, the signals we send, are deflected again
 We're still connected, but are we even friends?
 We fell in love when I was nineteen
 And I was staring at a screen

Entre la nuit, la nuit et l'aurore.
 Entre les voyants, les vivants et les morts.
 If this is heaven
 I need something more
 Just a place to be alone
 'Cause you're my home
Title: Re: Lambert's Cosine Law
Post by: vampares on 21/09/2014 05:51:36
I thought, I found a way to enter
 It's just a reflektor (It's just a reflektor)
 I thought, I found the connector
 It's just a reflektor (It's just a reflektor) Just a reflektor

It's just a reflektor (It's just a reflektor)

It's just a reflektor (It's just a reflektor)

It's just a reflektor (It's just a reflektor)

reflektor (Just a reflektor)
 (Just a reflektor) just a reflektor
 (Just a reflektor) just a reflektor (reflektor)
 (Just a reflektor)
 (Just a reflektor)
 (Just a reflektor, just a reflektor)
 (Just a reflektor, just a reflektor)

Just a reflection, of a reflection
 Of a reflection, of a reflection, of a reflection
 Will I see you on the other side? (Just a reflektor)
 We all got things to hide (Just a reflektor)
 It's just a reflection of a reflection
 Of a reflection, of a reflection, of a reflection
 Will I see you on the other side? (Just a reflektor)
 We all got things to hide (Just a reflektor)

Alright, let's go back
 Our song escapes, on little silver discs
 Our love is plastic, we'll break it to bits
 I want to break free, but will they break me?
 Down, down, down, don't mess around

I thought, I found a way to enter
 It's just a reflektor (It's just a reflektor)
 I thought, I found the connector
 It's just a reflektor (It's just a reflektor)

It's just a reflektor

It's just a reflektor (It's just a reflektor)

It's just a reflektor (It's just a reflektor)
 It's just a reflektor (It's just a reflektor)

Thought you would bring me to the resurrector
 Turns out it was just a reflektor (It's just a reflektor)
 Thought you would bring to me the resurrector
 Turns out it was just a reflektor (It's just a reflektor)
 Thought you would bring to me the resurrector
 Turns out you were just a reflektor
 It's just a reflektor (It's just a reflektor)
 (It's just a reflektor) Just a reflektor

Just a reflektor
 Just a reflektor
 Just a reflektor
 Will I see you on the other side?
 It's just a reflektor
 Will I see you on the other side? (reflektor)
 We all got things to hide (reflektor)
 Just a reflektor
 Will I see you on the other side?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/09/2014 15:01:53
For light the geometry at the event horizon no longer has a recognizable 3 dimensions. The outward path normal to the horizon now equals zero and this dimension is inaccessible. This does not mean that we only have a two dimensional environment across the surface of the horizon as gravity will still act on the photon. This leads to the conclusion that beyond the horizon we have a negative 3 dimensional space. If the spacetime is flattened at the horizon this can be the only conclusion. This ultimately leads to a breakdown in the physics. To pass through the horizon light would have to be two dimensional with no depth to the energy. Internally this then translates to negative energy. This was one of the reasons that Maxwell could not proceed from electromagnetism on to gravity. However his consideration of negative energy was nothing to do with black holes.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/09/2014 22:31:29
I am currently looking into this and whether it can relate at all to symmetry broken photons.

http://physics.aps.org/articles/v5/44
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/09/2014 00:50:20
If we consider gravity losing energy and mass over time and distance from source to dark matter and dark energy then at an infinite distance it will no longer exist as gravitation. The dark energy/matter can no longer operate as a repulsive force at infinity so must be a mainly localized phenomena and follow the same inverse square law as gravitation. The majority of this dark matter and energy would reside in the vicinity of the source of the gravitation. That is the galaxies that originated the gravitation. The gas cloud G2 would therefore be prevented from serious interaction with sag a* simply because of the higher concentration of dark matter/energy coincident with the accretion disk and surrounding environment.

NOTE: The amount of this mass loss is directly related to the spacetime density surrounding the mass generating the gravitational field so this would not even be noticeable in a terrestrial environment.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/10/2014 16:36:36
I am now of the view that mass loss may not be the answer. Gravity probe B results show the vortex around the earth as predicted by Einstein. This runs counter to the direction of angular momentum as if, like an induced magnetic field, gravitation is the interaction of an external field with the moving mass. The Higg's field imparts some of the mass to a particle. This relationship may be part of the answer. Energy may still be lost by gravitation but not sufficient to explain dark matter and dark energy. The voids between the galaxies are always being affected by gravitation however infinitesimally small the effect in those regions. This could over time lower the general mass density which in turn lowers spacetime density. I am looking at the falling slinky effect in order to determine a method of describing the density change in spacetime.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/10/2014 02:52:53
I now have an equation to determine the radius of the start of the accretion zone around dense objects, including black holes. The conclusions explain the failure of G2 to be consumed by sag a*. It also brings up the possibility that once a black hole forms nothing will ever fall into it. While this sounds like it must be wrong there are reasons for this conclusion. These involve non-violation of light speed and were calculated indirectly from the gravitational binding energy.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/10/2014 22:41:29
Gravitational binding energy U is given by U = (3*G*M^2)/(5r). Where G is the gravitational constant, M is the mass and r is the radius to the surface. THe mass must be considered of uniform density. Gravitational acceleration g can then be determined by g = (5*U)/(3*M*r). If we extend the radial distance and consider a drop in density within our new surface boundary then U decreases proportionally. This appears to indicate that density is a critical factor in gravitational calculations. This density can only be expressed through a variation in G which makes it frame dependent. This relates G not only to mass density but to spacetime density.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 07/10/2014 00:03:36
This all indicates that we are chasing a ghost by trying to find a constant value for G.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/10/2014 00:25:26
I am now starting work on a universal time dilation gradient with its reference point as the event horizon of a black hole. I will be able to use this gradient in further equations and possibly then link gravitation directly to maxwell's equations.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/10/2014 21:39:49
If we take GM as our starting point and find the fractional portion of the mass used in gravitational calculations we can then find the depth of distribution this would be around the surface. This then shows a relationship between the depth of this mass and the radius of the event horizon. The concentration of the mass then reaches a critical point at which the intensity of gravitation produces the energy required to overcome the escaping photons. This surface depth then relates directly to the radius of the horizon. It also shows just how weak the gravitational field is.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/10/2014 21:50:29
It could be that most of the gravitational energy in a mass is confined and only a small portion of the flux leaves the mass at its surface. This cannot be determined without an investigation of this surface depth at increasing densities in relation to gravitation forces. If this depth equals the radius at the horizon this would answer some questions. It may also confirm the frozen star hypothesis.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/10/2014 23:43:40
The speed of light can be determined by the equation 1/(G*50). This equation can be rearranged to find the exact value for G in the local frame. The factor of 50 relates to the fraction of mass across the surface found by using GM.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/10/2014 00:00:24
Therefore 1/50c = G.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/10/2014 10:24:58
A while ago whilst investigating the 2GM in Shwarszcchild's calculation I came across the factor of 0.02 which I knew was important but didn't know why. Well 0.02 = 1/50 and now I know why.  [8D]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/10/2014 16:05:54
Our G value is in seconds per meter so if we take the reciprocal 1/G this gives 50*c which may well be the speed of gravity. The speed is then 14,983,877.348 km/s. This would be why gravity cannot be detected easily. You would need to take 50 equally spaced measurements and look for a repeating pattern of vibrations that could not be explained any other way.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/10/2014 17:08:06
The electromagnetic spectrum could them be linked to gravitation by interference of the wavelength of the graviton with that of the photon. That is if the graviton exists.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/10/2014 00:06:47
Just for good measure I have attached a plot of a composite wave interaction with a standing wave.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/10/2014 21:33:16
Attached is a model representing the removal of the gravitational component from a compressed mass. The length contraction decreases to zero through the progression. The gravitational constant has been replaced by the 1/50 factor. This plot starts with a mass compressed within its event horizon radius. The length contraction dies off gradually until during the last 20% the rate of change increases dramatically.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/10/2014 22:18:49
The equation for our starting point is 3d17c18adfddfb0f8fe249228ba931af.gif whose units are cubic seconds per cubic meter. In other words 3 dimensional time in 3 dimensional space. This is a much better solution than using G.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/10/2014 23:49:28
Attached are two plots. One calculating g using the gravitational constant and the other using a factor of 1/50c. The question is this; is the factor of 50 actually a constant. We view gravity from an isolated bubble and have limited data on differing strength gravitational fields. This is one thing I am looking into. The other is gravitational interaction at the particle level.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/10/2014 01:33:09
Although the results match there is a real problem in using c in the denominator. Instead this should be L which then represents simply the distance traveled by light in 1 second. G is used as a unit conversion and can be calculated approximately by using hbar/(lP^2C^3). However we need a unit conversion that instead of producing G will produce the unit value of 1. So hbar/(lP^2C^3) requires such a factor before being added to our equation. The reciprocal of this equation will give us this value as a constant. Then it needs to be determined what this constant means.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/10/2014 02:21:04
So fingers crossed our equation for g should be (hbar*M)/(lP^2*c^2*L*r^2). Of course the Planck length would need to be recalculated using 1/50c in place of G to get an adjusted value but ignoring units.

Note: Changed g to G related to the 1/50c factor.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 02:20:24
I have not checked the results for the above equation for consistency yet. The point of interest is that the factor of 50 has disappeared. This suggests that there can be no energy loss in gravitation. So no link with dark energy/matter. Dark energy is entirely separate from gravitation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 03:52:13
Well now like Joyce's Finnegan's wake we go back to the start with:

-e8b40a892f462413b0c8d876775c483b.gif
-096e31371fcc75f544a6e1aa4dfc126c.gif

What about replacing G with 1/50c in this equation? The other point of note here is that we have 3 constants in the left hand term. That is G, Pl and Pm. It was shown that the factor of 50 can cancel out but in this case it would still be present, I think. This is the next project. To incorporate gravitation into the electromagnetic wave equation. In this way a formula for the affect on the wave over time by gravitation can be developed. Hopefully!
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 03:53:15
Then it will be back to the standing wave interaction.  [:0]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 04:30:18
One thing that comes out of this is that the radial distance from a source will be affected by length contraction which is not catered for in gravitational calculations such as that for g or escape velocity. For a distant observer this would be important but not an observer local to the frame that the radius refers to. The factor of 50 must vary as the radius varies so any function to adjust for change in apparent length must operate on this factor as well as the radius. This could be one way of integrating gravity with quantum mechanics.

Note: To achieve the integration the starting point to choose would be the imaginary surface of the event horizon where a fixed value exists.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 04:58:34
I have checked the equation (hbar*M)/(lP^2*c^2*L*r^2) and it was a terrible guesstimate. I will go over it again to find out where I went wrong.  [>:(]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 23:01:59
Right. L = the distance a photon travels in 1 second. c = the speed of light. lP = the Planck length. hbar is the reduced Planck constant. Our factor of 1 is then [50*L*c^3*lP^2]/hbar. This I have checked and it is near as damn it to 1. So this factor is then applied to M/[50*L*r^2] which is our calculation for g (gravitational acceleration).
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 23:24:30
That then gives us [M*c^3*lP^2]/[hbar*r^2] for our gravitational acceleration. What the units are I have no clue so any help would be appreciated. It may be totally invalid.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/10/2014 23:50:34
Since the factor of 50 has canceled what we can say now is there is no energy loss. This means that it is the field density alone that affects its strength. So as in the case of the electromagnetic field the gravitational field must have a force carrier. This will be the spin 2 boson.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/10/2014 09:23:19
Taking 2 candidate galaxies and their central supermassive black holes it should be possible to calculate a time dilation gradient between them. The unknowns are the masses of each black hole and the extent of the event horizons of each one. If these were known then our starting point at each of the galaxies would be the imaginary surfaces of the spherical horizons. This has to be imaginary due to any bulging caused by angular rotation. The effects of recessional velocity and any intervening masses would affect this gradient over time. It would be best to start with an idealized model with known variables. This would be made more difficult due to the velocities of the black holes. This causes the point at the horizons surface to be modified away from a fixed universal value. Isn't relativity fun?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/10/2014 09:36:16
The solution of course is to have an observer midway between the galaxies and moving relative to both in order to maintain an equal distance from each one. This then becomes our second fixed point and cancels the effects on each event horizon.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/10/2014 21:57:27
I am now going to attempt to put together a theory of quantum gravity. Wish me luck.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/10/2014 00:19:57
Well I thought nobody was paying attention so I'll come clean. Remember [M*c^3*lP^2]/[hbar*r^2]? Well the c^3 isn't really that. It has L tucked away which is a scalar. What it should be is [M*c^2*L*lP^2]/[hbar*r^2]. So we now have Mc^2 in our gravitational equation. Quantum gravity is easier to deal with at lower energies and includes too many infinite variables at higher energies at the Planck scale. What happens if we have energy implicit in our equation and an intimate link to the Planck dimensions?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/10/2014 00:31:08
We can think of c as being L/1. That is 1 light second. We then end up with (L/1)^2 standing in for c^2. This can be viewed as [L^2*lP^2]/1. Square area over time. Multiplying this result again by L gives us a reduced cubic area over time which includes an energy component implicitly by virtue of the mass in the numerator. This resolves to a spatial containment of energy reduced by a proportionality at the Planck scale. This is exactly what we need to start down the path of quantum gravity. By removing the need for G and replacing this with energy we can more easily integrate this with the wave equation and consequently electromagnetism.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/10/2014 23:03:38
Reviewing the equation [M*c^2*L*lP^2]/[hbar*r^2] can show us some important points. (L/1)^2 can also be viewed as (dL/dt)^2 where t cannot go below 1 as this would make the speed of light superluminal. So dt >= 1 and then dL <= L giving both time dilation and length contraction implicit within this function. Then M(dL/dt)^2 gives us a changing energy profile as the properties of the system change. Our L*lP^2 gives us a volume of space over which the change of properties occur. Having hbar as the denominator seems to indicate a Planck variant scale in changing frames. Which implies a constant Planck scale over which mass density changes. This is why relativity has not been resolved. The curvature has been viewed in the wrong way. So now we can separate the functions into [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)]. This is an energy against volume equation relating energy density to gravitation as both time dilation and length contraction vary. It is also directly connected to the effects of gravity on the photon.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 24/10/2014 01:09:01
Considering the equation [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)] it is apparent that r, the radial distance from the source, has to bear a proportionality to dL/dt as this is the distance from the source of gravitation and therefore relates to the strength of the field. The dilation and contraction must fit with experimental observation. In the case of length contraction this is difficult to achieve.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 24/10/2014 01:11:23
Once the proportionality is established this then forms the basis of the time dilation gradient between black hole event horizons. This will include a full model of the behavior of light between these fixed points.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 24/10/2014 23:03:59
There is one more pertinent point to make. To measure in a frame dependent way we would use the form [M(L/t)^2]*[(L*lP^2)/(hbar*r^2)] where L/t does not vary. So that local observers all read the same result. This is equivalent to the form GM/r^2. For observers measuring remote frames then the form is [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)].
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/10/2014 19:02:21
To separate out time dilation or length contraction we can vary time whilst keeping distance constant or vary distance whilst keeping time constant. I have attached two graphs showing this. For the time dilation graph the x axis starts at 1 second which is the local frame. Any point away from this shows the decrease in change over time. At the 2 second point, for instance, it will take 2 seconds for an action to happen that would take 1 second in the local frame. This can also be seen in the length contraction graph. The curves are not equivalent and operate differently. We cannot read off equivalent energy values from both graphs and use the time and distance values taken in our original equation as the energy would differ. The next step will be to develop the dilation/contraction equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/10/2014 18:31:28
To correct the equation we cannot only vary length or time whilst holding the other constant. We start with E = M(L/t)^2. This can be rearranged to give SQRT(E/M)*t = L. This graph is linear and energy is now constant. This is the time dilation gradient. The graph is attached.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/10/2014 18:33:11
Since we already removed the factor of 50 earlier this shows conservation of energy in gravitational interactions. NOTE: This is a graph of energy distribution over time and distance. It relates to the speed of energy flux.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/10/2014 21:14:32
Here again the local observer is at 1 second and the distance is L.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/10/2014 21:27:21
We can see from the graph that if we look at say the 2 second position on the graph and read off the distance we get twice L. This is not length contraction. The speed of light has not changed at all. What does this show us? Well energy MUST vary otherwise gravitation has no effect. The first two graphs separated out time dilation and length contraction and showed a difference in those curves with respect to energy. This means there is a factor missing. This could indicate a negative energy component. This would be our gravitational energy. Without this we cannot validate the effects of the gravitational force. Without it gravity doesn't exist.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/10/2014 21:44:00
I asked a question in the the Physics, Astronomy and Cosmology forum as to whether length contraction exists. The next derivation should prove that it doesn't. This is in fact an error that has caused a stall in relativity.If you look at the graph where length is held constant and time changes we see a curve that looks like an inverse square equation. This is absent when we hold time constant and vary length. I will post this another time.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/10/2014 01:15:07
We can rearrange the equation for g to get the time factor as SQRT([M*L^2]/g*[L*lP^2]/[hbar*r^2]). This graph is also linear and starts at L/1. This can be thought of as the speed of light at an infinite distance from any gravitational source. Moving horizontally right is equivalent to moving into an intensifying gravitational field where light is effectively slowed down as viewed by an observer at infinity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/10/2014 01:22:47
The time axis in the above graph will reach infinity at the event horizon. Any marked effects represented by the plot will only occur near massive dense objects. The attached plot of earth's g shows no noticeable difference on the time axis as this will be measured in nanoseconds and will be unobservable under normal conditions. This can be considered a linear relationship in a less intense gravitational field.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/10/2014 08:12:00
Kinetic energy is given by E = (1/2) mv^2. E is kinetic energy, m is the mass and v is velocity. We can derive momentum as p = √(2Em). The deBroglie equation for wavelength is λ = h/p where here λ is the wavelength, h is Planck's constant and p is momentum. This can be written as λ = h/√(2Em). Since we have already derived t from the gravity equation then we can also derive m. Then time and kinetic energy will be variables in the wave equation. This can be used to show the evolution of the wave under the influence of gravity. In which case m becomes the mass of the particle with r^2 indicating the particle radius squared. The value of g for the particle is most important. This is our way into quantum gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/10/2014 20:16:50
The most profitable investigation would be into the relationships between mass-energy, kinetic energy and time.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/10/2014 23:18:18
I am now in the position that I need to derive the mass equation.  This is to test an hypothesis that gravitation is merely a catalyst and not a force in its own right.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 31/10/2014 00:34:14
So we end up with M = 022cf3cf7cc30d53b6ced8b8682a3313.gif-1. Looking at it this way the g force is inherent in the mass with gravity as the catalyst.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 31/10/2014 01:19:17
Finally we can rearrange as M = g0c6cc1626df2bd5b3703d2de75ac6d03.gif-1. So now we only need modify g and r to find the mass contained within a radius that will produce a particular g force. We can attempt to apply this to a particle or to a black hole. There is a direct relationship to density inherent in the equation. This neglects time dilation and length contraction but it is straightforward to modify to take these into account. However the complexities rises with four independent variables.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 31/10/2014 01:51:26
Our mass equation can then be substituted into the momentum equation p = √(2Em) where L/t does matter as this has an effect due to both time dilation and length contraction. It can also be a way of showing the effects on kinetic energy. We have to be careful in the application of this formula as it is the mass energy that changes in order to have an effect on the kinetic energy. Which is the wrong way round. Unless we consider it a change in flux rate.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 31/10/2014 20:59:58
I think that gravity as a catalyst is unworkable for a variety of reasons.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 01/11/2014 01:36:43
To get energy Mc^2 we arrive at fc72c725d5aca2f490339c036bbfca38.gif-1. Which we can rearrange as Mc^2 = c6db32e5758711fef86612c1d2371b0d.gif-1. The deBroglie wavelength being h/p we should be able to use these equations to model the effect of gravity on particle waveforms. For momentum we get p = √(2kEg/c^2[{lP^2/r^2}{L/hbar}]^-1).
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 01/11/2014 03:18:47
Having the Planck area lP^2 in the equation is of interest.

http://en.wikipedia.org/wiki/Planck_length
"The Planck area, equal to the square of the Planck length, plays a role in black hole entropy. The value of this entropy, in units of the Boltzmann constant, is known to be given by A/4lP^2, where A is the area of the event horizon. The Planck area is the area by which a spherical black hole increases when the black hole swallows one bit of information, as was proven by Jacob Bekenstein."

The term lP^2/r^2 can therefore link our mass-energy to a density function that relates to the horizon black hole.

Another important point on this page is this.
"In doubly special relativity, the Planck length is observer-invariant."
So is length contraction valid or are space and time separate.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 02/11/2014 15:11:30
Upon reading further there are serious problems with double special relativity so I am going to ignore it for now. The energy equation is important as it relates energy to gravitation rather than mass. As the photon is massless this is the only way we can use the equation with the photon wave equation. I will be looking at Pete's relativistic mass page soon to see how it can all be combined.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/11/2014 23:08:29
We can finally rearrange the mass equation from M = g0c6cc1626df2bd5b3703d2de75ac6d03.gif-1 to M = g6f83b50cde111e699ce130a87d63a53d.gif to remove the reciprocal with units of joule second^2 metres^-2.

Correction the units are joule second metres^-2. And of course 1 (joule second) per (square metre) = 1 kg / s
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/11/2014 00:06:09
So now if we multiply this mass value by 1 second we get our kg value. Since our 1 second value relates to light speed. As time dilation increases our time value increases. Multiplying by the new value gives our increase in mass due to time dilation and increasing velocity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/11/2014 01:45:09
This of course gives us mass flow rate A.K.A mass flux. This is usually used in fluid dynamics although there is no reason not to use it in other ways.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/11/2014 00:30:03
The mass equation M = g6f83b50cde111e699ce130a87d63a53d.gif can be rearranged in the same way the energy equation was to become M = geda2b363d67f8bce7b72742c33d4bb94.gif. The term aca7705d5933d7248a073b1ecb9a6de7.gif gives us the number of Planck squares in our radial square area and therefore reduces the magnitude to a scalar Planck multiplier. If the value of r were 2lP this would equal the rs value of the event horizon of a Planck mass black hole. The scalar value then becomes 2. Interestingly this is the factor in the rs equation 2GM/c^2. To derive the Planck mass value from this expression would simply require finding the corresponding value for g.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/11/2014 00:43:24
To hold r at 2lP and simply increase g is the same as increasing mass within a set volume. Therefore increasing density. When we get to the point where instantaneous acceleration equals the speed of light we will have found a value of great interest. This will be the point of no return at which a singularity is inevitable.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/11/2014 23:07:00
We have kinetic energy ek = 5f6a292a60d11ac93192fece04c53090.gif and potential energy ep = -421685ad858d70073dbbaf837ed38d95.gif. These equations balance as energy is converted from potential to kinetic and visa versa. What is not taken into account here is the effect of time dilation due to a changing gravitational field. The derived equations above can however take this into account. Using energy instead of mass in these equations is the only way to proceed when dealing with massive or massless particles.
To verify length contraction an effect upon kinetic energy must be present. This must be equivalent to a loss of energy when viewed from a remote frame. I intend to show that there is no such effect upon the overall kinetic energy of an object moving through a gravitational field. As the kinetic energy reduces there is an equal amount of increase in the potential energy. This can be thought of as the kinetic energy being negative and the potential energy as positive. Kinetic energy becomes positive only when approaching a mass through its gravitational field. Which is why no force is felt. When accelerating outside of a significant gravitational field, a mass carries with it its own gravitational field and so does feel a force as it cannot be moving through its own field. Its kinetic energy is inherently negative.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 00:20:54
So we arrive at ep = ca2042ab86038480c6fef4e1b3c199a1.gif. We now need the same form for ek. Here we have m^2 s^-2 but with hbar having joule second units which signify angular momentum. So how do we square this with a kg unit?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 01:34:42
Well one answer from yahoo is:

https://answers.yahoo.com/question/index?qid=20110820184453AABGGvX
"Do you really mean m^2/s^2 and not m/s^2, which is just acceleration.

 But m^2/s^2 could mean many things. For example, during the recent nuclear disaster in Japan, the radiation dose rate received by workers and citizens was measured in Seiverts/hour. The unit of dose, the Seivert has the dimension m^2/s^2, which is equivalent to Joules (energy) per kg:

 Example: Energy(Joules) = force x distance (Nm) = mass x acceleration x distance (kg m^2/s^2)
 Energy per mass = Dose (Seivert) = mass x acceleration x distance / mass (m^2/s^2)"

So this could be thought of as energy per mass. What of the hbar in our denominator in joule seconds?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 01:50:29
Ignoring the hbar for now what of our energy equation ep = ca2042ab86038480c6fef4e1b3c199a1.gif. What use is it? We if we consider the g factor to be the gravitational acceleration at the surface indicated by the radius r the we can have an external ge value. This value will be the g force from an external mass acting upon the local mass in ep = ca2042ab86038480c6fef4e1b3c199a1.gif. We can determine the change in ep by using 19549f9a4c19d9f946b249d33540e623.gif. If g>ge then our potential energy is positive. if g<ge then our potential energy is negative. What happens when g=ge. This is the situation where the fields are said to cancel. This is equivalent to reducing the potential energy by cancelling g in the equation. So that zero gravity situations will decrease potential energy. Where does it go? Usually we would think of this as becoming kinetic energy. This is one puzzle whose solution will ease our way into a theory of quantum gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 02:49:09
Because we have a cancellation of g our kinetic energy must be internal. This indicates a low point in time dilation. So that within a hollow cavity at the centre of a mass time dilation will be at its lowest. Since all the forces of the outer mass cancel then the only g force present is that of any particle at the centre of the cavity. This bears out the theory that an event horizon must start at the centre of mass and work its way outwards. This also indicates that composite particles should merge in order for the cancellation to operate outward. If the particles were still individual then a full cancellation would not apply for all particles. Only a unit mass with a single gravitational force can cause such an inward collapse.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 04:05:12
Having sat and thought about this the equation 19549f9a4c19d9f946b249d33540e623.gif is not valid at all. While the situation in a hollow cavity would still apply this subtraction of the external value of ge in the above is incorrect.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/11/2014 23:11:38
The equation for Planck energy is SQRT([hbar*c^5]/G). This can be derived from the above as [g*h]/[pi*L]. This dispenses with the square root and the gravitational constant and returns the value in joules. Importantly the Schwarzschild radius of the Planck mass is inherent in the equation and so can be rearranged to find g. Our value for g is then 2.77943185E+51. This accelerates mass to superluminal velocities according to this result but does it. Not if we take length contraction into account. The speed of light is never actually violated. Therefore length contraction must exist. This also implies that kinetic energy has a different relationship to gravity than has been thought previously. When moving through a length contracted frame the locally viewed acceleration will appear faster than is apparent to an observer in a remote frame. This will only be noticeable near to a dense massive object with an intense gravitational field. Solving Einstein's field equations in this situation becomes a real challenge. Do we even need to?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/11/2014 00:29:00
The g force works out to be very approximately pi*c^6. What significance this has I have no idea. Except that it must relate to length contraction in intense gravitational fields.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/11/2014 01:17:47
A while ago I came across the work of Paul Marmet. The significance of his work has only become apparent to me very recently. He was an opponent of general relativity so was largely ignored by the mainstream. I hope to vindicate him not by proving general relativity wrong but by modifying it.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/11/2014 07:54:28
I have copied two of my posts from other threads.

1) I don't want to do that. I want an equation of the form 9c0982c9ae7f496f41433e6e4e02bdc8.gif in three dimensions. Here energy is not simple to describe. The components of mass, potential and kinetic energy interact with the gravitational field.

2) If we take the centre of gravity of a perfect sphere and have a plane running through it. We can then define x, y and z axes tilted so each is axis has the same angle to the plane. If we then set a path that when projected onto the intersection of pairs of axes is at 45 degrees all axes that describe the path perpendicular to the plane then change at the same rate. As a baseline for mapping the effects of rates of change this can map a straight line path. This can then be adapted for curved trajectories. Extending this path out to an imaginary spherical surface the mass within the surface can be defined to be of any size with a radius of choice. Comparisons are then easy to make against the baseline straight path. It would be interesting to see what effects we could model on the interchange of energy under various conditions. To include the electric, magnetic and gravitational fields. With an equivalent value for rate of change at equidistant points the effects on energy of multi-mass systems would be fairly straightforward. Just an idea.

Taken together these two ideas can ultimately produce equations of the type derived by Maxwell for the gravitational field. Very early in this thread a gravitational component replaced the permittivity/permeability factor in one of Maxwell's equations.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/11/2014 23:47:00
Just as g can be shown to be an intrinsic part of a mass equation can we include a factor describing the de Broglie wave equation itself? If so maybe then we can show exactly how particles behave in a changing gravitational field.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/11/2014 23:24:03
E0 = m0c^2 relates rest energy to invariant rest mass. We have derived ep = ca2042ab86038480c6fef4e1b3c199a1.gif for potential energy. What if we rearrange like this? ep/g = d9ef166fcf80212594bdf63bea4431af.gif. Here g is the gravitational acceleration AT THE SURFACE of a mass. It cannot be anywhere else. Here g is an acceleration in square seconds, c^2 is a squared velocity. WE can say that ep/g = ex where we do not know what x represents. We know it must represent a reduced mass and therefore a reduced energy but cannot be separated from the mass from which it was derived. However a change in the dimension of the radius of the mass will change the inherent value of g at its surface. This also means a change in density. If the radius increases so g decreases and the reduced mass value is larger. If the radius decreases then the value of the reduced mass gets smaller. So therefore the proportion of mass that is involved directly in the gravitational force then increases as is shown by other gravitational equations indirectly. This shows directly how density varies the gravitational force via a redistribution in the balance of mass involved in gravitational interactions.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/11/2014 23:27:55
The only 2 values that appear to change in the above equation are mass radius and g at the surface. Here lies the first problem. How do we decide on the radius of a particle? It isn't like a beach ball. In fact how do we really know what it is like or how it is distributed. The only way we can proceed now is via the wave equation that is well defined and experimentally verifiable. Can it be done? At this point I don't know.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/11/2014 00:01:10
What happens when the value of g = c^2? Then e0/g can be said to be similar to e0/c^2. However it is not an equality. It is an artificial modification. The gravitational acceleration becomes superluminal for a start. The points of interest are the resulting reduced mass term and the mass radius.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/11/2014 19:27:13
The basic problem with the mass and energy equations derived in this thread are their application. They can tell us nothing significant about the particle since the radius is uncertain and variations of mass/energy at that scale are too small to investigate. In the case of a macroscopic mass the equations neglect the nature of the mass as individual particles within molecules which combined together have individual interactions that would invalidate results. This is why the derivation of an equation with an intrinsic wave component is one of the only ways to proceed.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/11/2014 13:36:30
Can we incorporate the wave into the mass equation? Let's start with two equations. The first for kinetic energy and the second for the wavelength itself.

Here KE is kinetic energy, m is the mass and v its velocity.

KE = (1/2)mv^2


For the wave equation we have:

f = h/p

Where f is the frequency, h is Planck's constant and p is momentum. To incorporate kinetic energy into the equation the following steps are required.

KE = (1/2) mv^2

2KE = mv^2

2KEm = m^2v^2

2KEm = (mv)^2

Since momentum equals mv we can derive momentum to include kinetic energy using SQRT(2KEm). Then for the wavelength we have:

f = h/SQRT(2KEm)

We can never have zero kinetic energy because we always have zero point energy. Therefore KE has to be intrinsic to mass which means mass always has momentum. Only  for purposes of mathematical derivation can we use rest mass. Since mv would require velocity to be a numerator we would be multiplying velocity with hbar so no we cannot incorporate the wave equation into a mass equation. The same can be said for the energy equation. This indicates that the wave is merely an effect of the motion of the particle through space. Either in a straight line path or via angular momentum. From this we can reach the conclusion that because the wave is not intrinsic it can be directly affected by the gravitational field. Since the gravitational field will affect trajectory.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/11/2014 18:03:13
Now in a previous post we did see how a wave can be affected by gravitation. The plot is shown again here. This was arrived at by examining Maxwell's equations. It is not a verified equation by any means. What it does attempt to show is the shift in wavelength as a particle moves outward from gravitational field source. As a starting point this needs to be re-examined rigourously.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/11/2014 18:10:25
What the above plot does bring to mind are the discrete energy levels and integer wavelengths of the electron orbitals.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/11/2014 18:25:39
We can develop a 3 particle model to show gravitational interactions as vectors. We have to use 3 since we can describe a plane that all the particles line up with at any point in the evolution of the interactions. Any more particles cannot be assumed to sit on this moving plane. The inherent values of g for each particle can then be described as vectors in the system as it changes over time. Determining how each wave evolves during the interactions will be of interest. Exactly how do they behave?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/12/2014 03:03:02
A correction to one of the posts above. In the equation f = h/SQRT(2KEm) of course f is wrong as it is the frequency and not the wavelength. In case I confused everybody. It should be λ = h/SQRT(2KEm). If we hold mass as invariant then the kinetic energy determines a change in the wavelength. Since gravitation alters the kinetic energy this can be used to describe the effects of gravitation on the wave. More on this later.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 03:12:00
The Kaluza-Klein theory is described here:

http://en.wikipedia.org/wiki/Kaluza–Klein_theory

This is a scalar theory of gravitation. Interestingly from here:

http://en.wikipedia.org/wiki/Scalar_theories_of_gravitation

We find that:

"Kaluza–Klein theory involves the use of a scalar gravitational field in addition to the electromagnetic field potential  in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by Pascual Jordan."

The fact that this leads to a variable gravitational constant is of interest. Only one of the papers appears to have a translation. This theory has some interesting consequences and should be pursued vigorously in my opinion. I will be investigating this in conjunction with the work shown in this thread.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 03:28:37
For reasons I will explain when I have worked out the proof, mass can exceed light speed but only when approaching an event horizon. Within a defined region surrounding the horizon nothing will be visible. It is not that things disappear once the horizon is crossed. They will vanish BEFORE the horizon is reached. The innermost point of an accretion disk will mark the extent of the outer visible area. X-ray sources must then emanate from this region as they would not escape the region beyond this.

NOTE: It MAY be possible to achieve superluminal interstellar velocities but only if it is possible to shield against gravitation. This has to be considered with caution since it relies on propositions that are entirely without proof.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 22:17:12
I have discovered a relationship in the gravitational field density. This is shown in the attached graph. I will not be showing how this was derived at the moment as this has far reaching consequences if correct. I will be developing a set of equations around this initial equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 22:59:21
Contrary to what I may have posted earlier in this thread the gravitational field in fact does lose energy but the equation describing the rate of change is not a simple relationship. More energy is lost nearer the source than further away. In fact the field later regains some of the lost energy from somewhere. This is puzzling. Gravity well is an understatement of the situation. I can now derive the density variations outside the event horizon and the 'no light zone' before the start of the accretion disk..
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 23:01:57
If we consider the energy of the gravitational field as negative then this implies the field is becoming more positive as it moves away from the source. This does not mean that it will ever become a repulsive field but may explain the accumulation of dark energy due to energy conservation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/12/2014 23:39:28
The possibility of gravity giving rise to the dark matter/energy halos comes from the profile of energy loss. Starting low then reaching a peak and dying away again. This would produce such a halo effect with most of the dark material concentrated at the peak of energy loss. The dark matter/energy produced would be even weaker than the gravitational energy and would accumulate over time. A galaxy that has had the material stripped by an encounter with another galaxy could over time re-acquire its halo due to future energy loss from the gravitational field of the central black hole. These anomalies may be detectable and it may be fruitful to find such galaxies.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/12/2014 22:51:30
One consequence of the above hypothesis is that the electromagnetic field will lose energy in the same proportions as the gravitational field. This does not apply to the magnetic field itself which loses no energy. This makes sense as it circulates and would be unable to sustain circulation if energy was lost. So the electric portion of the field and the possibly photon itself lose this energy the further from the source the position of the field or particle is. I say possibly in the case of the photon as I just don't know for sure.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/12/2014 23:48:31
Gravitational lensing should be more pronounced at a set radial distance from the mass generating the gravitational field. This will be within the halo region around the mass at a set radial distance from its surface.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/12/2014 00:16:31
The other thing all this finally establishes in my opinion is that gravity does in fact travel at exactly light speed and itself undergoes dilation due to its interactions with the electromagnetic field. They affect each other proportionally. Like the charge of the proton and electron being the same while the mass differs the electromagnetic and gravitational field affect each other proportionally even though their energies are not equivalent. I have no idea how this works.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/12/2014 02:00:45
Finally we come to our equation for the wave. If we rearrange the mass equation to be M = bfce3b9ba37a3b47a1bd9a97bad0404f.gif we now have a time dilation component implicit to the function. This includes the velocity of the mass. Applying ae539dfcc999c28e25a0f3ae65c1de79.gif to the inverse light speed t/L also relates to the dilation of the photon in a gravitational field. However this form is concerned with velocity alone. Interestingly the square of the radius and the surface value of g also increase. This echoes the thinking of Paul Marmet that the Bohr radius must increase with velocity. The next step is to test the equation for the results it will produce to see if in fact it does reflect the real world experimental data.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/12/2014 17:40:25
The mass dilation equation has some of the fundamental constants incorporated into it. It would be interesting, although maybe not very informative in its present state, to try values at the Planck scale for radius and g. This can be done using the variation derived for the Planck mass black hole. Using this a fixed reference point can be set at the event horizon.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/12/2014 15:37:07
I have found an interesting papaer which may be of interest. I haven't read it yet. It is from December 2011 so is only 3 years old. The title is "Where is hbar Hiding in Entropic Gravity?". This is to do with the proposal of entropic gravity by Erik Verlinde which I also haven't reviewed. This is apparently a classical Newtonian gravity theory with origins in quantum mechanics. The link is:

http://arxiv.org/abs/1112.3078

Just a note from me. Is hbar hiding in the definition of mass as above?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 20/12/2014 16:59:19
One concrete proposal has come out of these investigations which I will need to provide the equations for eventually. For a black hole with the mass of the earth with rs set at around 1cm there is a special region which extends outward radially to a distance of 815.4 metres (Approx.).I am not entirely sure if light would be trapped within this zone, probably not. However any other tardyon mass WILL be trapped within this zone. So therefore the event horizon is not the danger zone for the possibility of escape. This already occurs further out. This may in effect prevent the measurement of the mass of black holes with any accuracy.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/12/2014 15:41:45
The results above had led onto an equation that includes one function for angular momentum and another for specific area. I am not sure what the specific area indicates as it is only indirectly related to specific volume. These equations when viewed with respect to the earlier equations in this thread should provide some new insights on the interaction of mass with gravity. I am working on this now and will post the details when the equations are complete.

EDIT: If a function of angular momentum can be used in the mass equation then this will allow the evolution of the wave due to gravitation to be an inherent property of mass. This may also become a way of describing the action of time dilation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/12/2014 18:22:34
In the equation below we can determine the distance traveled during an amount of time t when the gravitational acceleration equals g.
Δy = 1/2*g*t^2

If we then set g to equal

g = 2L/t^2

we can show that in one second due to cancellation of the following

1/2*2L/t^2*t^2

That we will have traveled L distance in one second. Since L is equal to 1 light second of distance this means that we will have reached light speed during this acceleration. Applying this to the parameter for earth gives us an exclusion zone around an earth sized black hole. This is the first step in deriving our new mass equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/12/2014 00:04:27
If we set g = 2L/t*dt we have

2L/t^2 = Gm/r^2

Re-arranging for r

r = SQRT(t*dtGm/2L)

If r = L then

L^2 = t*dtGm/2L

Re-arranging for m

m = 2*L^3/t*dt*G

Setting G equal to the approximation 1/50c

m = 2*L^3/t*dt*50*L/t

Restoring c

m = 100c*L^3/t*dt

And finally re-arranging

Volumetric acceleration
m/100c = L^3/t*dt

Cumec for volumetric flow is in the units m^3/s so here we have the potential volumetric acceleration of gravitation for the whole mass. We need to reconcile this with g at the surface that will be the next step.

The above is equivalent to the following simply formula.

Gm/2

Here the factor of 2 appears again. So if we let a equal this volumetric acceleration we arrive at:

a = Gm/2

EDIT: To cater for length contraction in 1 spatial dimension we would modify the equation thus:

m/100c = L^2*dL/t*dt

Now we can cater for the effects of both relativistic changes with respect to the potential volumetric acceleration.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/12/2014 01:09:56
The difficulty of reconciling gravitational acceleration with volumetric acceleration lies in the fact that while the first follows a linear geodesic the second represents an infinite number of radial directions in 3 dimensional space. These all emanate from the centre of gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/12/2014 16:46:22
So how to reconcile this. Well we saw that y = 1/2gt^2. If we re-express a = Gm/2 as a = Gm/(2r^2) we find the equivalent acceleration for the length y during a 1 second interval. What needs to be determined now is how an expression for angular momentum can be achieved.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/12/2014 02:03:59
For a general wave equation we can show the following progression:

m = g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]

P^2 = 2*Ke*g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]

λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

In the final wave equation the velocity v should be considered the only variable. It is a component of both Ke and gamma. Since we have components such as c^2, h and the Planck length squared it would be interesting to review equations containing these combinations to see if this gives us an insight into gravitation, mass and quantum mechanics.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/12/2014 03:17:29
As a starting point for an angular momentum formula we need to review the following.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html

Another useful reference is on wikipedia.

http://en.wikipedia.org/wiki/Azimuthal_quantum_number
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/12/2014 18:15:30
To develop our equation of angular momentum we must first review an earlier post.

"Before trying to link the gravitational field to Lambert's Cosine Law I need to take a detour. This starts with the unit sphere and the unit circle. Using the unit sphere and circle shows some interesting relationships and can be scaled up. This can then be used to describe both subatomic and macroscopic domains.

The circumference of the unit circle is 2*pi. To determine the angle of an arc around the circle whose arc length is equal to the radius we can use (1/2*pi)*360 which can be simplified to 7/44*360. This proportionality will become important when viewing interactions at differing scales and relates to wave frequency, length contraction and time dilation effects. The angle we have determined can be converted to radians to use in calculations.

It is interesting to note that the period of sin x is 2*pi. This can be utilized by considering forward motion and angular rotation as it relates to the unit sphere. The relationship between these two properties can describe the evolution of a wave and can be related directly to the gravitational field. When used it can be shown to show the underlying mechanism of the Pauli Exclusion Principle and the difference in energy levels required between electrons.

There are 3 directions of motion under consideration within this model. One motion is forward direction and is considered to be aligned with the poles of the sphere. The two other directions are angular. The first is around the equator and the second follows a longitudinal path intersecting both poles. The maximum unit of motion in unit time in the polar direction is equal to the unit sphere radius. The maximum unit of motion of the angular paths is 7/44*360 as stated above. If viewed at the Planck scale the angular components cannot reach this speed or none of us would be here. Therefore we can deduce that this dampening in angular momentum must be due to gravity which is what the current physical theories state.

If we follow this line of thinking through to its conclusion we can show that when considering the universe as a whole system light might get infinitesimally near to c but will never actually reach it as long as any gravitational field remains. I will demonstrate the reasons for this conclusion as I proceed."

The angle derived above becomes important only when the radius of an object is 1 Planck length. As this will only apply in some string theories we can disregard it. Not only because it relates to string theory but it relates to light speed which angular momentum should never reach. This will be discussed later.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/12/2014 15:48:42
In examining angular momentum it is not sensible to have Planck units. The best units to select would be nanometres and nanoseconds. Thus the speed of light can be represented by L*1 nanometre divided by 1 nanosecond. We can then apply a factor to L to determine a non-relativistic speed. Now we may have a problem as the angular momentum of a particle is L = r x mv. The two values for L, 1 light second versus L for angular momentum mus NOT be confused. L = r x mv is the cross product mv the linear momentum and r which is the point of rotation. This is best illustrated in the animation on wikipedia.

http://en.wikipedia.org/wiki/Angular_momentum
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 01/01/2015 00:12:17
I have decided to temporarily suspend posts to this thread until I have reviewed the following.

http://phys.org/news/2013-11-proton-radius-puzzle-quantum-gravity.html

http://arxiv.org/abs/1412.4515
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 02/01/2015 00:14:46
What if gravity or any other such force which is considered inverse square is not that at all but appears that way the further from the source and at macroscopic scales. If instead the law was 1/r^(1+1/n) and n started at an as yet undetermined value > 1 and approached 1 but only got there at infinity then we would have an entirely different situation microscopically where gravity starts as a 1/r law and then changes over time to become nearer and nearer to a 1/r^2 law. What if we started the value of n at 50 as an arbitrary assignment?

EDIT: This may validate the dark matter halo hypothesis.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/01/2015 02:42:28
In one of the papers by Roberto Onofrio the Schwarzschild radius of the elementary particles becomes boosted by 33 orders of magnitude. As light should still be trapped then the energy of the gravitational field should become boosted by an equivalent factor. So we could replace the value of 50 by 33. If we stay with 50 for now then the attached graph shows the change in gravitation with respect to an earth sized mass. The radius and mass are held constant while the power of the radial term moves from 1/r^(1+1/50) to 1/r^(1+1/1). At the left of the x axis the situation mimics a Planck mass situation where the whole mass can be treated as a single particle. At the right of the x axis we return to our 9.81 value of g where the mass is made of separate particles. The x axis has been graded so that the intervals between successive points are more closely spaced at the source and separate with movement to the right. This is the only way to achieve a sensible scale and illustrates the change from 1/r to 1/r^2 as a gradient.

EDIT: The x axis is derived by starting with a value of 50/50 at the left and ending at 50/1 at the right. This does not mean that this is valid in this scenario. It was a quick experiment with the values.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/01/2015 03:17:52
This would of course imply a decrease in energy of the gravitational field the further from the source it is. As stated above the intervals on the x axis were derived. If we remove the derivation then we get the attached plot which does show a very different profile altogether.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/01/2015 23:38:56
We can carry out a very rough guestimate on our first value by using the mass of the hydrogen atom and the mass of the earth. We then get a factor of Mh/Me where Mh is the mass of the hydrogen atom and Me is the mass of the earth. Our first value for g was 38913462.45 so we end up with (6.7e-28/5.97219e24)*5.97219e+24. Our result is 4.36557e-45 for g. However this is a 1/r relationship and does not describe an acceleration in this form so the actual acceleration will be much less. If we apply the same proportionality to the radius of the earth as we did to the mass we can use this in our next guestimate.

EDIT: We would need to obtain the square root value of the reduced radius as a correction. If this value does not match a reasonable radius for the hydrogen atom then this invalidates this method entirely. This will need to match the Bohr radius within a certain range of precision.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/01/2015 00:07:05
What we get is 2.67495e-23 metres which is 31 orders of magnitude larger than the Schwarzschild radius of the proton. Since Roberto Onofrio is suggesting a particle's Schwarzschild radius is boosted by 33 orders of magnitude this is a very surprising result and only 1 order of magnitude out.

CORRECTION: This should state a 2 order of magnitude difference.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/01/2015 00:19:30
The attached graph is a log/log plot of the evolution of the variation in the profile of the gravitational field in terms of induced acceleration. The values for mass and radius are still those of an earth sized mass and no scaling down has been attempted.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/01/2015 21:43:37
The scale at which the forces would be acting is the attometre. This is 10^-18 metres which is 5 orders of magnitude greater than the boosted proton Schwarzschild radius of the proton. Our scale could be defined from the boosted radius up to the currently accepted proton radius which is in the femtometre range 10^-15. The proton radius is around 0.84–0.87 fm in size. Can an equation be derived that shows a morphing of gravitation between the microscopic and macroscopic domains? This is the key question. It will not be like the previous guestimate.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 07/01/2015 03:38:51
The next step is the investigation of the gravitational coupling constant. Details can be found on wikipedia.

http://en.wikipedia.org/wiki/Gravitational_coupling_constant
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/01/2015 02:18:39
A short detour is worth taking before continuing. Consider a point P which is at an infinitesimal distance outside the event horizon of a black hole. At point P a photon is emitted in a direction directly away from the gravitational source. Since gravitation will slow light in its field and the the escape velocity is infinitesimally near c at point P then the emitted photon will not have the energy to achieve a velocity equaling c. Since it cannot reach the escape velocity it should therefore lose kinetic energy as it moves away from the source. The question that arise is does this mean that the photon is therefore trapped and will eventually stop and reverse direction. Since, without any other forces acting on the photon, it will only experience a constant velocity it cannot accelerate to escape the gravitational field. This is an important consideration when investigating the ergosphere surrounding a black hole. This is suggestive of a zone outside the horizon that can still potentially trap photons. Since it has been shown experimentally that light can be slowed substantially when moving through certain mediums it may be that the intensity of a gravitational field can mimic such a medium. Infalling photons that are not directed at the source may well fall into near-horizon orbits. Being perpendicular to the direction of the gravitational field may allow photons to reach a velocity infinitesimally close to c. However this may not be the case at all.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 09/01/2015 02:41:51
It is useful to read the following article.

http://news.harvard.edu/gazette/1999/02.18/light.html

The extremely low temperature of the medium will effectively transfer kinetic energy in the form of heat. This implies that the ergosphere around a black hole is acting like a bose-einstein condensate. This can come about via infalling matter achieving this state with low vibrational energy. This implies a uniform change in acceleration on the particulate matter within the ergosphere so that no heat is generated and no force is felt. Since kinetic energy is equivalent to a potential heat transfer the field, or something, must be taking up this heat. Either that or x-rays with enough energy are effectively escaping the ergosphere and taking this heat away. However, what if there is no matter close enough to the black hole to be falling into the ergosphere? In this case there will be no medium through which light can be effectively slowed down. In the case of the encounter of gas cloud G2 with sag A* it may simply be the case that the black hole was starved of matter with which to generate a sufficiently dense medium with which the gas would react.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/01/2015 18:25:25
Consider the whole surface of the horizon of a rotating black hole. Because of this rotation frame dragging will carry matter around the equatorial disk. Velocities will be high. However at the poles this velocity is absent and therefore makes it easier for matter to enter the black hole's event horizon at those regions. This could result in energy release in the form of photons in all directions. Some will fall into the black hole at the pole. Some will initially form eccentric orbits across the horizon while others will be propelled away from the poles. The jets away from the poles being intense will carry some matter with them in the form of maybe gas molecules or nuclei and separate electrons.

NOTE: Due to entanglement a proportion of the photons expelled via the jets could provide information on entangled photons falling into the black hole.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/01/2015 19:11:40
If we investigate path profiles around a rotating sphere it could be possible to construct a model of this scenario. This can then be related to smaller scales. Ultimately it can be applied to the particle itself. The following is a good point in the history of particle physics to start.

http://en.wikipedia.org/wiki/Spin_(physics)#History

"Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession).

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function."

Pauli's objections were overcome but should they have been? This will be investigated on the way to our wave interaction equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/01/2015 00:32:02
Another useful piece of history to review is this:

http://en.wikipedia.org/wiki/Llewellyn_Thomas

"Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician.[1] He is best known for his contributions to atomic physics, in particular:

Thomas precession, a correction to the spin-orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom.

The Thomas–Fermi model, a statistical model of the atom subsequently developed by Dirac and Weizsäcker, which later formed the basis of density functional theory.

Thomas collapse - effect in few-body physics, which corresponds to infinite value of the three body binding energy for zero-range potentials."

Particularly this:

"While on a Traveling Fellowship for the academic year 1925–1926 at Bohr's Institute in Copenhagen, he proposed Thomas precession in 1926, to explain the difference between predictions made by spin-orbit coupling theory and experimental observations."

A section on the Thomas interaction energy can be found here:

http://en.wikipedia.org/wiki/Spin–orbit_interaction

This includes a Lorentz factor for a moving particle. This is an important consideration for interacting wave equations, especially with regard to changes in a gravitational field. Time dilation can be related directly to this factor.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/01/2015 23:33:48
If we return to the poles of the black hole it should be true that a higher proportion of Hawking radiation will be generated here than elsewhere along the surface of the horizon. The interaction of particle wave ensembles away from the poles should be a good area to investigate for particle interactions with gravitation within an intense field. If entanglement is prominent here we may also be able to determine some aspects of the behavior of particles having fallen through the horizon in these polar regions. A study of the profiles of many polar jets should give valuable data in this regard.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 12/01/2015 06:01:45
Now if we want a balanced wave equation we first restore the factors of 2.

This:

λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Becomes:

λ = h/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Then we need h on the right hand side so:

λ/h = 1/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Inverting we get:

h/λ = SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Squaring:

(h/λ)^2 = 2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L]

Multiplying by 2pi:

2pi(h/λ)^2 = 2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

Finally:

2pi = (λ/h)^2*2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

We can now adjust g, r, wavelength or gamma*t but they must all balance in order to conserve the equality. This involves conservation laws somehow. Which ones, maybe all, I don't know yet. I may easily have made a mistake here so beware. This does not take into account the effects of an external g force but since 2pi is the circumference of the unit circle a multiplication be ge (external g) will result in either an expansion or contraction of radius depending upon where it is less than  or greater than 1. Being based upon metre units this is the correct pivot point.

NOTE: Since an increase in the unit radius would imply a slower spin rate then as the gravitational field intensifies this will correlate with time dilation. In this form the equation says nothing about a change in radius with increasing velocity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/01/2015 22:51:46
The reciprocal of momentum 1/p = λ/h. This may relate somehow?? to phonons or even the reciprocal lattice.

See crystal momentum:
http://en.wikipedia.org/wiki/Phonon

http://en.wikipedia.org/wiki/Reciprocal_lattice
"Simple cubic lattice[edit]

The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side 2pi/a (1/a in the crystallographer's definition). The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space."

How this could be incorporated is beyond me currently. It is simply noted here.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/01/2015 21:16:26
We have two interesting components to the unit circle equation. Ke = 1/2mv^2 and (λ/h)^2 = 1/p. Since mv = p we can define Ke as 1/2v(mv). That is 1/2v(p). By combining both we arrive at 1/2v(p)1/p which becomes 1/2v.

So we can now reformulate the equation as 2pi = 2*1/2v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L] from which we then get 2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]. So we now have the particle velocity included in our rearranged formula. This eliminates the kinetic energy altogether. We now have velocity over a time dilated interval. Now 4 components can change and are all related directly. These are the velocity of the particle, the change in time over which the velocity occurs (time dilation as viewed externally), the radius and associated g force at the surface defined by the radius. Do these need to balance to maintain a relationship with the 2pi? That I have not determined yet. However the equation is becoming simpler. It does suggest a tie in with black hole entropy with the 1/lP^2 factor.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/01/2015 23:30:44
If we look at the following sequence rearranging for v we can see that velocity is also contained within the gamma factor.

2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

2pi/v = g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

v/2pi = 1/g*1/r^2*1/h*c^2*[lP^2*L/gamma*t]

v(apparent) = 2pi*([L^3]/[gamma*t^3*g*h]*[lP^2]/[r^2])

Here v(apparent) is not necessarily the same as v in gamma. The values for earth of g and r can be plugged in to see what results are obtained at various velocities inside the gamma factor. This is the next step.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/01/2015 19:37:51
Well I think that I can safely say that the above equation is rubbish.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/01/2015 20:40:26
Having scrapped the rubbish in post 197 I can now investigate whether or not the earlier wave equation is valid. Firstly I want to point out a conclusion I have reached. If we have two perfect spheres identical in every respect; same radius, surface area, volume and mass, then on a direct line between the two centre's of gravity the force can be said to cancel as it is equivalent and opposite in direction. From this midpoint on the line between the centres we can describe a plane perpendicular to the line on which gravity will cancel at any point as the forces betwwen the masses will cancel. However the vectors will mean that any particle not exactly positioned on the adjoining line will tend to describe a straight line path along this plane until it reaches equilibrium again at the point where the plane meets the line. This is equivalent to an object dropped down a shaft through one perfect sphere that goes from one surface, through the centre of gravity and reverses direction at the opposite surface. Both of these scenarios can be said to be operating in a flat spacetime as long as the only forces present are the masses concerned.

The two sphere scenario raises another point. If we consider the sources to be two black hole an infinitesimal distance apart at the event horizons with enough distance to hold a particle between them without touching either horizon then because the gravitational force operates equivalently and in opposite directions there should be extreme length contraction which also implies extreme time dilation. So in the situation at the centre of the earth we should have a maxima of time dilation for the size of mass of the earth. This also indicates a compression of a central particle in all directions indicating that this is a proof that black holes do in fact form at the centre of a mass and work outwards.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/01/2015 23:26:03
If we increase the distance between the two black holes and consider the central mass to be emitting a synchronized sphere of photons then it should be possible to calculate the deformation of the photon sphere as it moves outwards. The line connecting the two centres of gravity will form a triangle along the line of the perpendicular plane and this would be an interesting situation to study as the spacetime will experience no curvature. The angle at the apex will form a special relationship between the strength of the gravitational field at a particular point and the effect upon the wavelength and frequency of the photons moving in those directions. The change in the waveform can then be calculated for other directions were the gradient of curvature increases.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 22/01/2015 23:44:37
A note on the two flat spacetimes. The shaft through the sphere is one dimansional being a line through the sphere. Movement away from a straight line path will drift into a curvature in the fabric of spacetime. In the case of the two perfect spheres we have a two dimensional flat spacetime. What is of interest and likely not possible is if we can determine a flat spacetime that is 3 dimensional. If such a spacetime can be determined then we will have either gravity shielding or anti-gravity. Like I just said I don't believe this is possible. We need a third derivative of spacetime that is not at the centre of a mass. This will be a point in spacetime and therefore zero dimensional unless there is some other way of achieving it.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 24/01/2015 23:24:08
In the model with the flat plane between two equivalent perfect spheres we can say that the centre of gravity of the whole system lies outside of either mass. The equilibrium point coincides with the point on the plane that is positioned on the line between the two individual centres of gravity of the perfect spheres. All particles coincident with the plane and initially stationary at points away from equilibrium will be drawn towards this equilibrium point by the combined gravitational forces and their vector directions.

We can then define situations in which particles that are not stationary may be drawn into orbits around the equilibrium point. This flat spacetime is a unique situation to model and removes the complexity of dealing with curvature of the geometry. This is an ideal model with which to examine the change in the wave function due to the influence of gravitation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/01/2015 19:10:31
We can introduce something akin to the uncertainty principle into our two perfect sphere model. If we start with an orbit perpendicular to the plane between the masses that passes directly through the equilibrium point this will be our point of uncertainty. At the point of equilibrium in a perfect orbital path there are now two paths the orbit can take. It can either continue around the original mass or go into a figure of eight orbit around the second mass. As all forces are equal at this point there is a degree of uncertainty here. This is in effect a quantum state and binary in nature.

NOTE: An intriguing third option is that the particle continues on the plane away from both masses on its flat spacetime. This now becomes a three choice situation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/01/2015 19:33:18
If we consider the third choice for the orbital path, then any particle following this type of path will act in a similar manner to a jet expelled from the pole of a black hole. The difference is that instead of a directed jet we get a distribution along a plat plane. Does this have anything in common with the relativistic jet?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/01/2015 02:22:40
Since a particle that is initially at rest on the plane will tend to move towards equilibrium in a straight line path then we can set two out of 3 dimensions to have zero rate of change. This then produces a scalar value for the gravitational force and is analogous to the path of the particle falling down the shaft through a perfect sphere running from one surface to the opposite surface and passing through the centre of gravity. This one dimensional path still has a direction along the plane and a magnitude. The lack of curvature in the spacetime simplifies the change in wavelength of the particle. The slight complication arises due to the vector directions of the forces of the masses above and below the plane. Since these are equal we can sum them since we already have the direction of the vector of the particle.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/01/2015 02:33:53
One reason why it is important to determine how waves may be affected by gravitation is linked to the Penrose Interpretation which is described here:

http://en.wikipedia.org/wiki/Penrose_interpretation
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/01/2015 00:32:57
A little speculation now. As the density of the gravitational field increases it is my assertion that not only does the remote observer see an object slowing down but it actually does. This is because the increase in density acts against the acceleration due to gravitation. Once inside the ergosphere this density is likely to also trap light. This then prevents acceleration from violating the speed of light as objects approach a black hole. This will also mean that objects disappear upon entering the ergosphere. A similar situation will occur when approaching light speed since unlike the photon tardyons have non zero rest mass that increases relativistically. Also the speed at which the particles will be traveling, being relativistic, will mimic an increase in density of the gravitational fileds of distant objects in the particles vicinity. This will become more pronounced when in the vicinity of a massive object. The attached image, which was posted previously shows the results of earlier calculations of this. At that time I had put this aside but now feel more confident in this assertion.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/02/2015 00:41:50
Returning to the flat planar spacetime we can consider the effects of a simple harmonic oscillation about the equilibrium point as a start in determining the change in the wave function. The problem is in finding how this oscillation behaves in such circumstances.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/02/2015 21:30:50
Two possibilities can be considered for the effect upon the oscillation. It can either be elongated in the direction of both sources or it can be contracted by the effect of the sources. Since length contraction is assumed in a gravitational field it may be best to start with the assumption of a contraction or flattening of the oscillation. This will leave the particle flattened along the plane of the flat spacetime. In which case the effect of the resulting pressure may reduce the energy flux and be the cause of any time dilation. This is worthy of further investigation and may be a fruitful way to proceed in describing wave evolution away from equilibrium.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/02/2015 02:39:13
If the oscillation is directed mostly in the direction of the plane this is equivalent to the freedom of movement perpendicular to the direction of a gravitational field. The constraint in this case is created by field cancellation and not due to proximity to the surface of a gravitational mass. This then becomes a case of special interest.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/02/2015 00:22:51
Before proceeding with an investigation of gravity's effect on the wave it would be useful to examine the attached graph. This shows the change in wavelength over speed for the electron. The values have not been rigorously checked but this gives a good starting point. Relativistic mass has been taken into account but no gamma factor has been applied. The wavelength shortens with velocity and therefore the frequency/energy increases exponentially indicating the increase in mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/02/2015 01:31:35
If we were to include not only the gamma function, but also the derived mass equation with an inherent value for g, then as the relativistic mass increases the internal gravitation will also increase and have an effect on the internal functioning of the particle. The rate of change of energy flux will be effectively slowed down within the particle radius.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/02/2015 23:33:54
If we return to our two perfect sphere model and consider a particle on the plane of our flat spacetime then we can show something interesting. No matter what starting position we take away from equilibrium there is a second plane that is perpendicular to the flat spacetime plane that runs through the centres of gravity of both masses. On this plane lie the vectors from the centres of gravity to the starting position that indicate the strength of the gravitational acceleration of each mass. If we label these g1 and g2 respectively then we can say that g1 = g2 as all forces are equivalent at every point in the flat spacetime. We can use this perpendicular plane mathematically to determine the vector magnitude of the combined forces at the starting point. If we consider this perpendicular plane to be in the z direction then we can consider the motion of the particle to be moving on this plane as well as that of the flatspacetime plane. Knowing this we can ignore the flat plane to simplify the mathematics, the x and y values being zero. This can now be considered a two dimensional problem in the z plane.

This may also simplify the integration of the wave function.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/02/2015 23:39:00
It is said that the spin of a particle can neither be speeded up or slowed down. This is like saying that light always travels at c. Yet we know that light slows down in a gravitational field due to time dilation. Taking this into consideration we can say that spin must also be modified by gravitation otherwise dilated observers would see an increase in particle spin. This is how we must view the wave function when it is affected by gravitation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/02/2015 23:28:01
As well as the path through the flat spacetime we can consider another path. This is described by the equation v = SQRT(Gm/r) and would normally represent the velocity required to maintain a circular orbit at a particular altitude. In the scenario with two masses this is not the case simply because of the interactions of the fields and how they will affect each other. This is of particular interest in relation to the behavior of the wave function and can be investigated later.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 12/02/2015 19:47:12
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement

Time is removed from the equation due to the constraints involved, so long as there is no violation of those constraints. It is not so straight forward in the case of the flat spacetime plane sitting between the identical masses as all points on the plane are not equivalent. However the case of the single circular orbit is a good starting point in the use of virtual displacements and a way to derive equations of displacements in time for circular orbits around one or both of our two masses perpendicular to the flat plane and passing through the equilibrium point.

EDIT: This is useful because there will be no change in the wave function in this type of balanced system. When a modification introduces change we can then describe the effects on the wave.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 12/02/2015 23:58:59
The coulomb force constant 1/08d4b3c3fb2605943d8d8a59acd08dab.gif is similar to the factor in the original derivation of the Maxwell equation in the early posts of this thread. In that derivation 1/fefdcd9c3807378ab2ae0a72566a5194.gif was the value used. If we want the electric field incorporated in our mass we must derive a new equation.

We start with the equations:

E = kQ/r^2

g = Gm/r^2

Here k is the Coulomb force constant and Q is the charge. So we now have electric and gravitational equation. What can we do with them. If we consider the mass to charge ratio m/Q and the charge to mass ration Q/m we can now derive a new equation.

E/Q = k/r^2

g/m = G/r^2

These can be further re-arranged:

1/Q = k/(Er^2)

1/m = G/(gr^2)

If we want charge to mass this then becomes:

Q = (Er^2)/k

Giving:

Q/m = [(Er^2)/k] / [G/(gr^2)]

Q/m = [(Er^2)/k] * [(gr^2)/G]

If we now want a mass equation we first re-arrange as:

m/Q = k/(Er^2) * G/(gr^2)

And finally our mass equation is:

m = kQ/(Er^2) * G/(gr^2)

Now we have two components, the product of which gives our mass. The Electric field and its charge and the gravitational field and the acceleration at the surface. We now have united the electric and gravitational fields within the mass equation from which we can derive the magnetic component. With a means to derive the electromagnetic field we can now unite this with quantum mechanics. My knowledge of quantum mechanics is currently sorely lacking so I will be investigating simple effects on the wave equation until I can fill in the gaps in my knowledge.

EDIT: I made a correction to the value for k
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 00:11:21
Now this shows exactly why the wavelength affects energy and therefore mass, the electric and gravitational fields being intrinsic to the wave equation.
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 13/02/2015 04:28:34
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement
I don't understand why you're talking about virtual displacements here. The only place I know of where they're used is in analytical mechanics, i.e. Lagrangian mechanics.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 17:01:35
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement
I don't understand why you're talking about virtual displacements here. The only place I know of where they're used is in analytical mechanics, i.e. Lagrangian mechanics.

All will become as clear as mud eventually. I am just trying something out. If it fails I will come back and eat humble pie m'lud.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 20:43:21
Particles have spin. Not strictly the same as angular momentum but they are considered to be in motion internally of an undetermined kind. (Physics has models for this but no definitive answer.) If we consider the two components of the electric and gravitational fields in the above equation with respect to this internal motion what does it tell us? Are the fields also in motion and if not why not? If they move are they oriented in the same or an opposing way? If they are static is this because of opposing forces? The strengths of these two fields differ enormously. It is hard to reconcile a static nature due to opposition when they are unequal in strength. I do not intend to provide any definitive answers to these questions, it would likely be impossible. However, these kinds of consideration need to be kept in mind.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 22:12:29
We can re re-examine the equation v = SQRT(Gm/r) for a circular orbit around a single perfect sphere containing a uniform mass density. If instead of velocity we start in a circular orbit and then start applying an infinitesimal acceleration our orbit must expand with this increase in kinetic energy. Mush like an increase in energy of the electron moves it into a higher orbital. If we continue increasing the acceleration we will describe a spiral trajectory away from the central mass. This can be said to be similar to the vortex described by Einstein. This first infinitesimal increase in velocity, while different to a virtual displacement, can be useful in a number of ways. As the electron moves further out from the nucleus its wavelength changes. This will be the same situation with our orbit around our single mass. At any point we can determine how the wave equation is affected. We require less effort in this regard than in escaping in a straight line path as we start with all points in the orbit perpendicular to the gravitational field making it easier to move. Therefore any exchanges of energy will be trivial.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 22:45:48
Now here we have a conundrum. We are increasing velocity and yet to maintain a higher orbit we need a decrease in velocity due to the inverse square nature of the gravitational force. Otherwise we continue moving away from the source. In order to rest in a higher orbit we will need to manage a controlled and precise deceleration. This throws up some issues about the electron as the field around a particle SHOULD also be inverse square. So what puts the breaks on the electron?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 22:53:10
Well the electron must undergo a very short range and short lived acceleration of just the right amount of energy to shift to the correct orbital. This has to be precise and so does not involve uncertainty although this in no way invalidates the uncertainty principle. The position and momentum can still not be determine as it is a scalar determinacy.

EDIT: This scenario is at the heart of the issue of determining change in the wave equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/02/2015 23:29:27
If we rewrite v = SQRT(Gm/r) as s/t = SQRT(Gm/r) then we can eliminate the square root (s/t)^2 = Gm/r. This can be restated as (s/t)(s/t) = Gm/r. If we move one of our distances (s) to the other side it becomes (1/t)(s/t) = Gm/r * 1/s. The value of 1/s then relates to the tangent to the orbit that moves us through an angle to a higher radial position. So we have s/t^2 = Gm/r * 1/s. By multiplying both sides by 1/t we then have an equation for jerk s/t^3 = Gm/r * 1/s * 1/t. Of course we really need to be looking at ds, d^3t and dt in these in this case.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/02/2015 00:07:28
Of course if we did not want to maintain our perfectly circular orbit we could simply remove the accelerating force and allow an elliptical orbit to develop. This leads to the thought that maybe electron orbitals have a lot in common with elliptical orbits.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/02/2015 01:23:20
If we consider we have an acceleration (a) and a jerk (j) our final equation forms become:

a = Gm/dr * 1/ds

j = Gm/dr * 1/ds * 1/dt

With our final velocity after deceleration :

v = SQRT(Gm/r)

We also have an adjusted mass equation:

m = kQ/(Er^2) * G/(gr^2)

This differs from what is usually understood as mass as the electric field and charge components have been added in. So this is a composite mass equation. An investigation of substituting this into the velocity equation would be interesting.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/02/2015 02:01:52
If we consider a large central mass with a smaller mass in orbit and both perfect spheres with a unifomr mass density we can say something about our velocity equation. Perpendicular to the field of the larger central mass we have a gradient of velocities that describe concentric circular orbit. Throughout the volume of the smaller mass these orbital profiles have differing effects from the nearest to the furthest away. At the point in the smaller mass that is nearest to the central body the velocity is at its peak speed and dies away at the surface furthest away. If the centre of gravity of this smaller mass passes exactly along the orbit at the correct velocity then those portions of the mass further away are traveling at a speed that would take a smaller mass out of the orbit. At the nearest point of the smaller mass to the larger body the speed is lower than needed to maintain an orbit at that altidude and a smaller mass at that point would tend to fall out of orbit towards the larger body. Therefore an induced angular momentum of rotation could be induced by such an imbalance. The point nearest the larger mass would be retarded in motion whilst the furthest point will advance in its motion. This is without taking into account any time dilation. Even in elliptical orbits this would be true.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/02/2015 02:26:47
It is interesting to note that there are already equilibrium points known as the Lagrange points. At these points in the 3 body situation objects will maintain an orbit and even a small deviation from equilibrium will be corrected to bring a mass back to equilibrium.

Details can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/lagpt.html


EDIT: Equivalent points should be present throughout the universe due to the interaction of the gravitational fields of galaxies.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 14/02/2015 22:25:21
In moving away from a stable circular obit we need an acceleration. However, acceleration tells us nothing about the mass that is moving. It tells us nothing about the force that needs to be applied to accelerate the mass. Different sized masses can not be moved away from equilibrium at exactly the same rate of acceleration by applying the same force. Different forces are needed. This says something about the propagation of force through mass. When considering free falling objects of differing masses in the same gravitational field this is not the case. It is always the same force. This indicates an action at the particle level right through a mass and acting simultaneously.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 17/02/2015 01:00:15
New thought.

x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)]

Don't know where this leads yet. I need to look at some particle data. Not sure what x will actually mean.

Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/02/2015 01:26:19
The two fundamental equations for the de Broglie wave are:

c6a6eb61fd9c6c913da73b3642ca147d.gif= h/SQRT(2*Ke*m)

5470b9993b5d776db89f25ac7cfff3a1.gif= Ke/h

In both equations the energy is kinetic represented here by Ke. This is energy added to the system. Kinetic energy itself has the formula Ke = (1/2)mv^2. We saw earlier that momentum p = mv and that c6a6eb61fd9c6c913da73b3642ca147d.gif= h/p. The momentum is the motion of mass though space and the kinetic energy is a result of this motion. If we consider time dilation and its effects upon motion over time we should be able to derive equations that describe the changes in wavelength, frequency and kinetic energy in any gravitational field. What we also need is the effect on the particles electromagnetic field.

As I continue this thread I will be introducing relativistic mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 19/02/2015 01:19:10
A short digression before I progress this thread. Hawking radiation is due to pairs of particles, a particle and it's anti particle. One of these falls through the horizon whilst the other escapes. In the case of an anti particle falling through the horizon it should, if possible, be able to annihilate with a particle in the interior. If the particle falls in, however, then the universe loses mass. The particle will simply become part of the singularity. This may not be true, I don't know. However, whether a black hole grows or shrinks has profound consequences. I will be following up on this next.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 19/02/2015 01:30:35
When plotting change of wavelength of the photon that escapes just outside a black hole horizon there may be a linear relationship between frequency and the speed of the photon as viewed from infinity. The graph of this situation is attached.

With the frequency held constant and the speed of the photon plotted as viewed from infinity we have a linear plot. If we were to add a gamma factor, which I believe is incorrect, the photon will lose all its energy at infinity.

This plot brings to mind the Hubble data. If we consider that the black holes at the centres of galaxies are constantly changing, either shrinking or growing, then the shift that they cause in photon wavelength may also be changing in as yet undetected way. This may be regular across all such black holes and may follow a law of its own. In which case the expansion of the universe may be biased away from its true rate.

It all depends upon how these black holes behave. The black hole sag A* may be able to help answer such questions. In the end the red shift may be only very marginally affected by such black holes. Only time will tell.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 19/02/2015 01:38:15
For comparison I have attached the plot with the gamma factor.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/02/2015 01:15:50
If we re-examine the factors kQ/(Er^2) + G/(gr^2) these can be re-arranged as:

kQg/(Egr^2) + GE/(Egr^2)

This is an interesting set of relationships. The charge is paired with gravitational acceleration and the Electric field is related to the gravitational constant. Since the charge is involved in the attraction and repulsion forces and g is a consequence of the attractive force of gravitation this is worth further investigation. It is also of note that the denominator has the electric field and gravitational acceleration as factors. The units still need to be sorted for validity and this is what shall be undertaken next. This is not a mass equation by any means. It is something entirely different. If this is valid the value should be determined by unit cancellation. What we have left will guide the rest of the investigation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/02/2015 04:01:33
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 21/02/2015 17:03:20
If we consider an individual particle interacting with a perfect sphere we can then say that at the surface of the sphere the particle will experience a force represented by Sg. For a perfect sphere this will apply at any point on its surface. We can then say that the total potential around that surface is Sg*4*pi*r^2. We could then calculate the surface area the sphere would have if contained within its Schwarzschild radius rs. We can then define this as As. The surface of then sphere at normal density is A = 4*pi*r^2. If we consider this as a ratio we can say As/A is a starting point with A/As representing the magnitude of A with respect to As. Can we then say that Sg*A/As represents the true increase in force?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/02/2015 02:18:54
Looking again at kQg/(Egr^2) + GE/(Egr^2) what does the numerator kQg + GE tell us? We have Coulomb's constant incorporating the speed of light, magnetic permeability and electric permittivity. This produces a product with the charge and the gravitational acceleration. Added to this is the electrostatic field strength as a product with the gravitational constant. For a particle such as the proton, what values do we get? Do these balance in any way? That is the next test.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/02/2015 15:49:25
Enough outrageous speculation. Let's get down to business. If we consider the wave as circular rather than sinusoidal we can describe the total velocity by (2*pi*1/4*w*f)/t. Here w represents wavelength and f represents frequency. This is an arbitrary choice of variables. So now we have an angular velocity over time. However if we look at the Lagrangian bd5aabbac9bc3e5a0bc6cd4eae2fe473.gif this is unsuitable as the velocity is not angular. As the wave moves through space it is stretched and so there needs to be a combination of both angular and forward velocity to produce a combined velocity whose momentum can be used in a modified Lagrangian. Tying the two velocities together in this manner and with the frequency held constant we can show how the system will evolve when viewed by a remote observer. This also ties the time dilation directly to the internal actions of the particle. We can thus show the evolution of of the change in the rate of change of a system moving into a gravitational field.

NOTE: This is why a flat spacetime is crucial to the development of these equations.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/02/2015 16:29:46
It should not be assumed that the forward velocity and angular velocity change in unison. Maybe for the photon this can be true but this exercise is not concerned with photons. We cannot examine the effects of gravitation if these velocities are exactly proportional.
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 27/02/2015 16:52:21
Jeff! What in the world are you doing in this thread? It's as if you're having a very long drawn out conversation with yourself. What gives?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 27/02/2015 18:37:33
I am just using it to put ideas down. Don't worry I haven't gone bonkers.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/02/2015 00:39:01
If z is the axis of the path of the particle we can derive the equation:

514b00fedf907ba3ea467f653965f452.gif
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 28/02/2015 01:26:23
Okay. You remind me or yor_on when you do this.  [^]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/02/2015 01:37:52
So I'm in good company then  [;)]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/02/2015 04:45:02
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.

Interestingly a reduced mass formulation would be better for this equation. Especially when considering the effects of gravitation.

http://en.wikipedia.org/wiki/Reduced_mass

The equation then becomes x = [kQ/(Er^2) * G/(gr^2)] / [kQ/(Er^2) + G/(gr^2)]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/02/2015 20:02:31
There is something interesting we can do with the gravitational acceleration equation. This equation is:

g = Gm/r^2

This is like using a reduced mass term as we have neglected the second mass. If we rearrange this equation we can arrive at:

r^2/G = m/g

We know that force is F = ma and as g is an acceleration we can form F = mg. If we rearrange to relect this we get:

r^2G = mg

But r^2G does not equal mg as the form r^2/G = m/g represented a proportionality.

This can be illustrated via

1/2 = 2/4

1*2 <> 2*4

1*2*x = 2*4

In this case it is easy to determine that the missing factor is 2^2

However if we investigate the units of both sides of the gravitational acceleration equation something interesting can be found.

G units = m^3 kg^-1 s^-2 cubic metres per kilogram second squared

left side units = m^5 kg^-1 s^-2

mg units = m kg s^2 metres per second squared

missing units kg^2 m^-4 is surface density squared

So we can make the equations equal with the right value for surface density squared as:

r^2Gd^2 = mg

Where d is the surface density squared. The value of d here is undetermined. However, calculating its value is very important. This indicates that it is only the surface density and not the mass as a whole that generates the force of gravity. This also explains the event horizon surface area relationship to entropy. The remaining gravitational energy is internal to the mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 01/03/2015 17:06:49
So if the force is related to surface density, then as we increase the radius away from the surface of the gravitating object what density are we then relating to? It must still be the surface density of the gravitating mass but as it dies away in an inverse-square manner can this tell us anything about the density of the gravitational field at points away from the source? In other words, can we quantize gravitation?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 01/03/2015 23:29:35
It may then be shown that surface density is:

d = SQRT([mg]/[r^2G])

When we expand the radius this can describe two situations. Firstly the whole mass can assume a lower density within the volume described by the new radius. Secondly we describe a point away from the surface of the mass that leaves density unaffected. The mass retains its original radius value. Both situations are equivalent. Therefore this equation should also describe the field density at any point away from the surface of the source. This can only be true for a perfectly spherical object with uniform density. This is suggestive of the quatization of the gravitational field.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 02/03/2015 07:33:39
While it would be useful to determine if d is a constant in this situation it would not help with determining anything relating to a black hole. It is escape velocity that we need to relate to density to determine if quantization can explain the effects on light. There would need to be a reformulation of the escape velocity equation to relate it to the same surface density. Both situations could then be examined. Along with the relationship between gravitational acceleration and escape velocity. The next step is the reformulation of the escape velocity equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/03/2015 11:34:56
Before attempting to reformulate the equation for escape velocity it should be noted that these equations have not been independently verified for validity. It may well be wrong to include surface density in this manner.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/03/2015 01:57:37
WE have escape velocity as Ve = SQRT([2GM]/r). We can first remove the square root as in Ve^2 = [2GM]/r. Kinetic energy Ke = 1/2mv^2 and we already have Ve^2. Ir rearranged we have (1/2)Ve^2 = [GM]/r. If we multiply both sides by M we get (1/2)MVe^2 = [GM^2]/r. Force F = M/a so to get the force we need an acceleration to be able to proceed. This now becomes more complex. For now we can simply view this as a derivation of the kinetic energy of our escape velocity. The square of the mass on the right hand side echoes the square of the surface density to some extent.

EDIT: This is a strange equation in reality because we are not considering a mass moving away from a source but the source itself. Which can't be moving away from itself. So what exactly does this formulation of kinetic energy actually represent? Well if both masses were equivalent it could be viewed as the kinetic energy required to separate them at the required escape velocity. I am very unsure about this one.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/03/2015 02:14:39
Can we then say that the kinetic energy required to separate any two masses is GMm/r?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/03/2015 17:40:14
WE can view this in a different way and similar to the way the last equation was derived. From (1/2)Ve^2 = [GM]/r we can rearrange as [(1/2)Ve^2]/M = G/r. Then taking the same step as in [(1/2)Ve^2]*M = G*r the units necessary on the right hand side are now kg^2 m^-1. The units kg m^-1 actually represent linear mass density but without the square of the kg unit. I have no idea what kg^2 m^-1 represents.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/03/2015 01:09:15
We can eliminate the mass from the left hand side and include linear mass density as:
(1/2)Ve^2 = G*D*r

Where the d in the previous equation represents surface mass density and here D represents linear mass density. So rather than an energy equation we now again have a velocity equation. We can then restore a more original form.

Ve = SQRT(G*D*r)

The other equation

d = SQRT([mg]/[r^2G])

can now be reformulated.

d^2 = [mg]/[r^2G]

1/g = m/[r^2d^2G]

So we have

g = [r^2d^2G]/m

And

Ve = SQRT(G*D*r)

Here the main point of interest is the combination of the gravitational constant with the radius and density in both equations. Both the value for g and Ve will be with respect to the mass surface. These may be only applicable to the perfect spherical mass.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/03/2015 01:50:16
Working through the equation for surface mass density we find the following. At any point on a perfect sphere with roughly the mass of the earth only 0.36 % of the total mass can be considered to be having any gravitational effect.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/03/2015 02:18:09
Of interest in regard to all this are the items on this list of astronomical anomalies.

http://www.technologyreview.com/view/414539/the-puzzle-of-astronomys-unexplained-anomalies/

EDIT: Could this be due to an incoherence in the gravitation fields of masses? This brings us back to Lambert's Cosine Law strangely enough.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/03/2015 21:23:57
A correction to the escape velocity equarion.

Ve = SQRT(G*D*r)

Should be

Ve = SQRT([D*G*r]/[(1/2)M])

And the density equation becomes

D = [(1/2)Ve^2M]/[GR]

EDIT: Here the units are kg^2 m^-2. The density relationship here is not straight forward. Here kinetic energy is related to the density and makes sense when considering this is derived from escape velocity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 11/03/2015 20:15:22
Having gone back and thought about the shift in wavelength then length contraction has to be proportional to the length of the wave as calculated from a remote frame. This proportionality is possibly direct but maybe indirect due to a difference in the gradient of the change of each. Time dilation is then related to an inherent twist in spacetime due to the gravitational field. This will only be observable in the vicinity of extremely dense objects and when strong enough will result in frame dragging.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/03/2015 22:32:24
The following equation has been derived to try to determine the minimum mass for a stable black hole.

g = c^2/[2rs+L/262993404a7f720f18beb5a525f009d5d.gif]

However some caution is necessary. The kinematic equation for distance traveled due to free fall is:

d = vit + (1/2)at^2

vi is the initial velocity, t is elapsed time, a is acceleration and d is the displacement. Any object having an initial velocity when far from a black hole may ultimately acquire superluminal velocity before reached the event horizon due to its initial velocity. If instead of the Chandrasekhar limit we use a value of 3 solar masses we find that the acceleration falls below c. In order for this to be the minimum black hole mass it needs a proof of non-superluminal speed before the event horizon. Otherwise physics breaks down exactly where it shouldn't.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/03/2015 23:48:25
A paper on maximum netron star mass can be found here:

http://arxiv.org/abs/1307.3995

This is pertinent to the above equation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/03/2015 06:01:32
The equation

d = vit + (1/2)at^2

will function perfectly well in a gravitational field such as the earth's. When it comes to the region near to a black hole things are radically different. The amount of change in the displacement increases more rapidly. This results in the elongation or spaghettification of matter as it approaches ever closer to the horizon. Smaller and smaller increments of time are then required to determine the actual displacement along the path of the in-falling matter. It is therefore more sensible to assume a point particle along this path. The time increments are still necessary but no consideration need be taken of the effects along a mass made up of multiple particles.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/03/2015 04:30:34
A note on something I concluded whilst considering the speed of gravity. In order for gravity to operate at c the equation for the energy of the gravitational field should be Mc2G3. Where M is the mass, c is the speed of light and G is the gravitational constant. The energy is then 30 orders of magnitude less than the overall energy of the mass. This is in line with the difference in strength between the electromagnetic force and the gravitational force.

EDIT: Note that Mc2G3 is NOT an energy equation. It is an example of the magnitude of difference between the forces. With this equation you end up with 11 spatial dimensions.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/03/2015 14:58:58
If we consider a slightly different equation GMt where t is time then G has the units m^3, kg^-1 and s^-2. We can cancel the units of kg^1 with the mass and s^-2 becomes s when canceled with the time parameter. We now have units of cubic metres per second which is flow rate through a volume of space. The GM would normally be converted to an acceleration via division by r^2. For a mass the size of the earth this gives 3.98574405E+14 m^3/s. The question is how do we interpret this flow rate? It is not due to the rotation of the earth as angular momentum is not included and neither is an angular velocity. What is always there is particle spin.

What we have here is a reduced mass term via G. So we are not considering a 100% flow rate. At the particle level this may indicate is that a proportion of spin angular momentum is responsible for generating the gravitational field. This can be considered as a twisting field at the lowest level source.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/03/2015 15:06:05
This proportionality raises other questions. Why is there such a proportionality at the particle level? Can we determine a relationship between G and the elementary particles involved? Does this indicate that not all particles are involved in generating the field? I do not have the knowledge of the standard model to carry this further. If any can find this useful and wishes to pursue this they have permission to use the ideas as long as credit to the source is given. Then again no one may find these ideas valid in which case nothing is lost.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/03/2015 15:17:01
A final point on this is that if spin angular momentum is slowed by time dilation then the force of gravity generated by a smaller mass in the field of a much larger mass is reduced. The larger mass will be almost unaffected. For the speed of gravity to vary in proportion to the speed of light in a gravitational field them this should be true.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/03/2015 17:11:42
To proceed further it would be highly advantageous to read this page through completely.

http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/04/2015 17:46:48
On the gravitational constant. A constant value of between 48 and 50 should allow the calculation of the constant without the circularity involved in using the Planck values. This is because the gravitational constant is itself used in determining these values. An equation of the form [r^2/(NMct)]V^2 is therefore required. Where N takes the place of the constant (48-50). If we introduce relativistic gamma into this equation we can then determine the coordinate change in the gravitational constant that will relate to the effects of time dilation and length contraction. The value of r used as a the numerator is the unit normal vector. To use gamm we first reformulate the equation as [r^2/(NMc)](s^2/t^3) and then gamma is applied to (s^2/t^3). The derivation of this equation will require some further explanation. Its form was initially derived to preserve the units of the calculated value of G. The equation itself leads to some interesting consequences.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/04/2015 21:00:12
Considering [r^2/(NMc)](s^2/t^3) s = 1 metre, t = 1 second and M = 1 kilogram. The gamma function normally has v^2/c^2 but here we take the escape velocity Ve which makes this functional become Ve^2/c^2. The escape velocity Ve will only ever reach c at the event horizon and this is regardless of the size of mass. Using this form of the function we can calculate the coordinate value of G at any point away from the event horizon. To apply this to a black hole we first must find the radial distance of the black hole that gives the required value of Ve and plug this back into the equation to find the coordinate value of G. This in turn can gives us a coordinate value for the gravitational acceleration g at points outside the horizon. So in this way we can calculate how time dilation affects this coordinate acceleration. This explains why outside observers should see objects slow down when approaching an event horizon in strictly mathematical terms.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 05/04/2015 03:55:44
It should be noted that these coordinate equations are only valid for the Schwarzschild metric, That is a non-rotating and uncharged black hole.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/04/2015 21:31:50
The attached graph is a tentative attempt to plot the coordinate value of G, the gravitational constant, from infinity to the event horizon of a black hole. This is not a verified equation by any means but is interesting none the less.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/04/2015 01:21:22
I have seen it stated succinctly elsewhere that "Energy is the timelike component of the four momentum". So can we derive coordinate Lagrangians from a point near to the value at infinity to a point very close to the event horizon. This will require a coordinate kinetic energy and a coordinate potential energy. The potential energy we can derive using the coordinate value of G. Now it is time to look at coordinate kinetic energy.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 18/04/2015 22:37:00
The relationship between G and the Planck values can be expressed by:

2148970feb3a30106dbaa07eb023842e.gif

This correlates with the expected entropy of a black hole. That is the cube of the Planck length divided by the Planck mass times the Planck time will give a value for G. However this is circular because you need G to derive the Planck units themselves. What came first the chicken or the egg?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/04/2015 22:06:01
If instead we use L instead of b15a3315a69ce382beea50cbaa779f80.gif and t instead of 5c2423d9bb6800593fddead13d1534d3.gif
we have 57e288a237fb8102bfe96408da639cb7.gif. With t set at 1 second and L at 299792458 m we can simplify as 3f1b2222c04b1f441db845920e3de839.gif. To find m we use 3f2356b370cd1be487d6204412b887cb.gif. The attached graph shows the coordinate acceleration towards the event horizon viewed from infinity for a mass of m. The value of m is 4.03726E+35 kg.

The pink line shows a standard calculation of acceleration and the blue line the coordinate value. The turning point in the blue line shows the point at which the observed object falling into the black hole appears to slow down.

The y axis shows acceleration m/s-1 and the x axis radial distance from the event horizon. [Correction]The y axis shows acceleration m/s-2 and the x axis radial distance from the centre of gravity.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/04/2015 22:50:08
To really appreciate what this means please read the following.


http://www.google.co.uk/url?url=http://genesismission.jpl.nasa.gov/gm2/mission/pdf/Giantstars.pdf&rct=j&q=&esrc=s&sa=U&ei=u1s9VeKTGcL4atibgMAL&ved=0CBsQFjAB&sig2=FVpc1EFdKKfTZIo8srjslw&usg=AFQjCNHVd9qJvEk-oK6wJRBUlnbXhZ-B2Q
 (http://www.google.co.uk/url?url=http://genesismission.jpl.nasa.gov/gm2/mission/pdf/Giantstars.pdf&rct=j&q=&esrc=s&sa=U&ei=u1s9VeKTGcL4atibgMAL&ved=0CBsQFjAB&sig2=FVpc1EFdKKfTZIo8srjslw&usg=AFQjCNHVd9qJvEk-oK6wJRBUlnbXhZ-B2Q)

If true then galactic sized black holes are the only ones that can exist and must have been formed during the early stages of the universe and were responsible for the formation of galaxies. The amounts of mass involved can not exist as ordinary stars and must be a consequence of the slowdown of expansion following the inflationary period.

EDIT: A better source is:

http://www.space.com/858-study-stars-size-limit.html (http://www.space.com/858-study-stars-size-limit.html)
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 30/04/2015 01:08:49
The derivative of GM/r^2 is 2GM/r^3 which will show the rate at which g is changing at varying radial distances. This derivative is also used in Malcolm S Longair's book Galaxy Formation Second Edition. I am about to buy this book to pursue this further. I have attached a graph of 2GM/r^3 for the proposed minimum mass black hole.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 02/05/2015 05:43:37
The compressibility of matter is the crucial point in this investigation. This relates to gravitational collapse.

http://en.wikipedia.org/wiki/Gravitational_collapse
[Gravitational collapse is the inward fall of an astronomical object due to the influence of its own gravity which tends to draw the object toward its center of mass. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body acting in the opposite direction. If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained (the exception being a black hole).]

The crucial sentence is "If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained".

The key thing is to plot all potential stages of equilibrium for a variety of mass sizes.

Another crucial point is this.

"According to Einstein's theory, for even larger stars, above the Landau-Oppenheimer-Volkoff limit, also known as the Tolman–Oppenheimer–Volkoff limit (roughly double the mass of our Sun) no known form of cold matter can provide the force needed to oppose gravity in a new dynamical equilibrium. Hence, the collapse continues with nothing to stop it."

The Tolman–Oppenheimer–Volkoff limit is then the key to determining if black holes of 3 solar masses can actually form.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 02/05/2015 06:07:32
In discussion of the Tolman–Oppenheimer–Volkoff limit it is instructivbe to note this:

http://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit (http://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit)

[In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7 solar masses, [2][3] which was less than the Chandrasekhar limit for white dwarfs. Taking account of the strong nuclear repulsion forces between neutrons, modern work leads to considerably higher estimates, in the range from approximately 1.5 to 3.0 solar masses.[1] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known. The mass of PSR J0348+0432, 2.01±0.04 solar masses puts a lower bound on TOV limit.]

where "the equations of state for extremely dense matter are not well known".

Discussion of the uncertainty can be found here:

http://en.wikipedia.org/wiki/QCD_matter#Phase_diagram (http://en.wikipedia.org/wiki/QCD_matter#Phase_diagram)

"The phase diagram of quark matter is not well known, either experimentally or theoretically."

This then opens the debate on lower black hole mass limit.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/05/2015 10:11:32
After having read a post in another forum I am going to assume something rather odd. That all particles travel at the speed of light. This is a mathematical device only as NO they don't all travel that fast. However what it does do is make all things equal. The speed of light is also considered to be the speed of gravity. So why not just use the photon? Well the photon is massless. If this assumption is also combined with massive particles then we may learn something from a mathematical derivation. It can always be corrected later to show realistic values.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/05/2015 12:42:57
For the purposes of this investigation relativistic mass has to be ignored. The object generating the gravitational field will be a Planck mass Schwarzschild metric black hole. So that we can disregard angular momentum.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/05/2015 12:54:02
If we take c=G=1 then a photon orbital becomes 2M which is at the horizon of the black hole. If we were to direct a constant directed light source so that the photons in the orbital increased over time there would come a point where the gravitation generated by the light would start to cancel with the gravitation immediately inside the horizon such that Ve would fall below c. This will become important later.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/05/2015 20:58:43
One thing that it would be useful to do would be to determine the forces active in the internal cavity of a mass. I don't think this has been attempted experimentally underground. What it would be nice to be able to do is this.

63f08a7b28f21c74535a4fd7a25e987d.gif

However, since these are vectors , at the centre of gravity they are said to cancel so a summation is not possible. A method is needed to determine the action of opposing forces that does not simply assume cancellation. The effect on particles will not be zero. Something must happen due to the outward attractive force that applies in all directions. This is not a trivial exercise nor is it a worthless one. It has implications for the examination of extreme gravitational sources.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/05/2015 23:14:29
Going back to the coordinate value of G we can first investigate the relationship between surface area and entropy. If we take the equation fb2b4faef787e41d98ab472054a04d6e.gif then we can find a value of g that will give the area of the surface of the event horizon. We can then substitute the value of coordinate G that would apply at the horizon. In calculating coordinate G it was found that G does not tend to zero as the radius approaches rs. This indicates that the size of a black hole indicated by the range of the apparent horizon when viewed by a remote observer will be smaller than expected. This could be one explanation of why Sag A* did not consume the G2 gas cloud. If the black hole is actually larger than we think but compressed into a smaller area due to spacetime compression then the tidal forces will be lower than expected.

EDIT: Of course as well as the coordinate value of G we also need to substitute the coordinate value of g.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 13/06/2015 15:17:33
Going back to the idea of all masses moving at light speed we then have an equation for kinetic energy of the form d80c0869605cdd5f19dd6464a729cce0.gif. This is obviously neglecting relativistic mass. How is this useful? That is the next step in the analysis.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/06/2015 11:27:42
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency. This was verfied experimentally via Compton scattering and the equation 0d3c0ae7cb5f27f74ddf111fe479f64d.gif. Here c6a6eb61fd9c6c913da73b3642ca147d.gif is the incident wavelength 96ca2e3e3e7dabc48ee7a1d0d06b2067.gif is the scattered wavelength and 2554a2bb846cffd697389e5dc8912759.gif is the scattering angle. Now 6e4315b8b985bb1cebf8fb992dd0e4dd.gif where m is the particle mass and c is the speed of light. What was shown above was (1/2)mc^2. How does this relate? That will be answered next.

EDIT: 9776ff4be626811b4e2dad37d08e929a.gif is the Compton wavelength of the target particle.
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 28/06/2015 12:45:55
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. f43d56f4d269389e3158d69322b9d577.gif was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 28/06/2015 15:17:11
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. f43d56f4d269389e3158d69322b9d577.gif was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.

Thanks for that clarification Pete. It makes things clearer for the reader.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/07/2015 12:28:41
I was going to progress this thread on the theme of light speed particles. However I wish to post an equation without any derivation. This has come from various conclusions which I will get to later. The equation is this.

3a0b9329966dbdf30e82b10a2aa0cc24.gif

Here F is force, mP is the Planck mass, me is the mass of the electron, rs is the Schwarzschild radius of the Plank mass and r is the radial distance of the electron from the event horizon.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/07/2015 12:46:56
Please note: Before anyone does dimensional analysis and complains that this is not a force equation, there is a parameter missing.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 25/07/2015 13:02:36
With the missing parameter (tP) Planck time we can derive the following definite integral.

d6377222ab9950956994b1858279146e.gif
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/07/2015 20:44:58
The equation for F2 is not the same as for gravitational potential energy which is of the form:

ea246cdf364c4a24cec3d37f96b2ee4e.gif

The reason for this is the derivation used to arrive at the F2 form. The major difference is in the power of the radius r.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 26/07/2015 23:57:13
If we take the equation for a circular orbit

0a721ee760f19669ad7d291ba33b637d.gif

we can derive the following

3b67b29fe9088c057e67c9e6528364f9.gif

multiplying both sides by m gives

8838ab89e97af24d174a15c1f84dfa4c.gif

dividing both sides by 2

fb6dbae0376611367e7493ea274fdba4.gif

simplifying gives

ed0711ab9d21f917ceb6c37d80e421da.gif

This results in the total combined kinetic energy of the orbit. Under the right conditions and with the right modifications could this be a way to quantize the gravitational field?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 29/07/2015 22:30:28
Some of the subtleties and difficulties involved in the quantization process can be seen on the following page.

http://edition-open-access.de/sources/5/25/ (http://edition-open-access.de/sources/5/25/)
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2015 21:59:26
In partial relativistic terms we can start with be905d71a1af157cb287b23035db23de.gif. However we need to modify this to be a3d796b85aea56846659103ebf9ea6f4.gif where M is much larger than m.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 03/08/2015 22:10:15
From there we can reformulate first as d03777f2d6071b7c62c8aebb7d6b845f.gif and then as b048880e921439d40230f57a54d10081.gif.

NOTE: This only applies to the Schwarzschild metric and circular orbits.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/08/2015 00:04:26
When the orbit is coincident with the horizon of the black hole v can take on only two values. Zero and infinite. Whilst infinite is the expected value, zero should also be given due consideration. If we look at the situation where maximum kinetic energy is given by 0686df72ff4b90a1aa0b96ce6d246774.gif and then take v to be half light speed in the orbital equation this then equates to a momentum of 742568d55e925f46f19870b53bc9a11f.gif. If considering the Planck mass a velocity of 1/2c would give as twice the Schwarzschild radius. Therefore in a distance of 2 Planck lengths the velocity has to either change to infinite or zero. In my personal opinion zero is a better bet. Since light speed is 1 Planck length in 1 Planck time. This gives a particles 2 Planck lengths in which to come to a stop. Since it would have to be traveling 1/2 a Planck length in 1 Planck time this seems more reasonable than achieving an infinite number of Planck lengths.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 04/08/2015 00:12:18
The crucial point to consider is the transition between r at 2rs and r at 1.5rs.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 01:15:08
There are two types of kinetic energy that can be considered in this model. The energy required to maintain a circular orbit at a particular distance and the energy required to achieve escape velocity at a particular distance. Looking at escape velocity it is interesting to look at the magnitude of change in the field perpendicular to the surface but working towards the surface from a remote radial distance. This way discrete differences can be shown by the relationship of (r+1)^2/r^2 where the units are in integer increments of the objects radius. The graph of this function, which applies to any size of mass, is attached. It starts at 20 radial distances and ends at the objects surface.

Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 14:33:18
If we take b048880e921439d40230f57a54d10081.gif and modify this to be a74b951ce26e7577c63d36063f2841b8.gif we can remove the addition of rs and we arrive at 4686434e5358fd27987347b900ad7a34.gif. The Schwarzschild radius calculation itself is 3a1912b039e44adc516fbee8d280e149.gif so that with our equation we ar dealing with half the Schwarzschild radius. That is 1 Planck length. So adding this to rs gives us an orbital velocity of c at 1.5rs. This means that, for a Planck mass black hole, particles with rest mass can only maintain a steady orbit at twice the radius at a velocity of (1/2)c. This also means that photons can ONLY orbit at 1.5rs. Of course the scales are so small that particles would actually have to be point sources. It reality this would be an impossible situation. As a model though it may yield some useful results.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 15:44:55
If we consider the photon's wave it is best viewed as a geodesic with an oscillation about the direction of its path. This is because moving to a higher or lower radial distance would destroy the orbit. This is an idealized situation. The length of the orbit along its straight line geodesic path is 11e182d4c29bdf5bad4882f4809ff77e.gif where r is 3 Planck lengths or 1.5rs. The peak of the wave occurs over an extended period of time and will take an astronomical number of orbital periods to occur and so for all intents and purposes we can view the photon as traveling a perfectly straight line geodesic. This APPEARS to imply that at Planck scales the photon can be viewed as having zero energy. This is a strange concept as it also implies no wave function. At these scales it is as if the photon is only a particle. This makes sense however as the photon is not an infinitesimal point.

When viewed in this light the weakness of the gravitational field becomes less important at the Planck scale. It all depends upon how small the unit vectors for the extent of the instantaneous action of the force carrier of gravitation. If the vectors are at a scale where the energy of the photon is infinitesimally small then the action of gravity will have a high magnitude compared to the photon. This will also be a function of force carrier density.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 16:06:09
Now we can see the crux of the problem. Since the strength of gravity is so much smaller than that of the electromagnetic field the wavelength should be longer. This implies an astronomically high density of gravitons to affect 1 photon. Since at the Planck scale the photon appears to have zero energy then this has to be true of the graviton. We could state that fa29718bb536eae2bd29b41acf74abdd.gif is the relationship that needs to be satisfied. Where G stands for the graviton and gamma for the photon. At the Planck scale x cannot be a stand in for energy. So what is x?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 16:10:08
This could mean a different probability density profile to either the photon or any particle with rest mass. That is that the graviton is smeared over a larger area than other particles.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 08/08/2015 21:05:57
It could be worthwhile to investigate Causal Fermion Systems with regard to this model. Especially the states of the Dirac sea.

https://en.wikipedia.org/wiki/Causal_fermion_system (https://en.wikipedia.org/wiki/Causal_fermion_system)
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2015 00:45:12
To map a set of equal incremental increases in orbital magnitude we find a sequence of 1/2rs, rs, 2rs, 4rs, 8rs, 16rs etc. Each of these increments show an equal increase in orbital velocity. This profile is much different to that of the force in the direction of the gravitational field which is inverse square.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2015 20:05:42
There is something I need to correct here. After having run calculations I have found that it is a radial distance of 3rs that has an orbital velocity of 1/2c. At 2rs the velocity is 70% the speed of light. Light speed is still at 1.5rs.

I am currently looking for a well defined equation for relativistic escape velocity. Once I have this then I will be trying to determine the relationship between the two. If any.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 10/08/2015 20:25:32
For the orbital equation the relativistic mass terms were removed. It would be useful to apply this to escape velocity. Then the correlation between the two can be found.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/08/2015 13:59:58
I have attached a graph of orbital velocity and escape speed at set radial distances from a black hole. Escape speed is so named as it is not a vector with direction. Any direction that does not intercept the horizon will do. We can simply choose an arbitrary direction that is normal to the surface of the horizon.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/08/2015 14:26:45
The attached graph shows the difference between the magnitude of the two velocities and therefore the kinetic energy required in the perpendicular and normal directions. At a point 2 radial distances from the source these are equal. This is a radial distance of some significance.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/08/2015 16:04:26
This difference in kinetic energy is of the form bf28637ac8d3407e8bc0020eef113fff.gif
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 15/08/2015 20:17:16
Quantization should be attempted at the point of convergence of kinetic energies. This is not possible at the Planck scale so the next step is determining the best mass size to use in this process.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/08/2015 14:56:43
Since the Planck scale of a Planck mass black hole allows no stable orbit for light the mass size chosen should be to a scale that does allow a light speed orbit. This should be the smallest mass for which this occurs so that force carrier quanta can be determined. This will be a function of spatial separation of force carriers due to the inverse square nature of the gravitational field. It will also be a function of force carrier density.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 16/08/2015 23:20:54
Could this zero kinetic energy cross over relate to the UV Fixed Point.

https://en.wikipedia.org/wiki/Ultraviolet_fixed_point
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 17/08/2015 00:27:50
Could this zero kinetic energy cross over relate to the UV Fixed Point.

https://en.wikipedia.org/wiki/Ultraviolet_fixed_point
I can't believe it. You're still talking to yourself, Jeff? I didn't know that you were still doing this after all this time.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 11:31:05
I have the best arguments with myself  [;D]
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 11:36:43
To correct the final form of the equation we have 10e3ebc86c54ec3980e5f64750d8e42d.gif
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 12:42:39
Now for some speculation. One problem has always been the loss of gravitational energy over time. How can a mass generate gravitational energy and not have an accumulating loss of energy over time? What if we had graviton pairs appearing out of the vacuum where one graviton returns to the vacuum whilst the other interacts with mass and takes some of its kinetic energy? Ultimately, there should be a mechanism for this virtual graviton to return to the vacuum. This way we are only dealing with the loss of kinetic energy and gravitation is no longer inherent to mass. Just like the photon is not inherent to mass, but a separate and distinct particle. This graviton pair production mechanism then echoes the process of hawking radiation.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 13:35:50
If we consider the inflationary period as a type of runaway expansion we can then propose something else. That gravitation only became aparent during the radiation-dominated era.

https://en.wikipedia.org/wiki/Radiation-dominated_era

The interaction of vacuum graviton pairs with photons and neutrinos then acted as a brake on inflation. The intensity of the interaction would then have partly contributed to the production of the CMBR via extraction of kinetic energy from highly energetic photons.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 13:50:47
To sink gravitons back to the vacuum we have only two choices. Infinity or the centre of gravity of an object. The most realistic option is the centre of gravity. This implies that black holes, as well as being entropy sinks are also gravity sinks. The proposed fossil gravity field outside the event horizon of a black hole may be a consequence of this process.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 13:54:52
This sinking of gravity inside a black hole, over cosmological timescales, may be the trigger to converting a black hole to an equivalent white hole. Thus a big bang.
Title: Re: Lambert's Cosine Law
Post by: PmbPhy on 23/08/2015 19:13:38
I have the best arguments with myself  [;D]
Does it help you at all?
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 23/08/2015 19:53:18
I have the best arguments with myself  [;D]
Does it help you at all?

Lol. I was joking Pete.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/09/2015 12:59:19
If it is considered that gravitation originates in the vacuum then what happens when two gravitons interact? The attached image shows this possible interaction as a Feynman diagram using gluon symbols as placeholders for the graviton. the two original intersecting gravitons swap paths at the intersection and draw one of a pair of vacuum gravitons out and in the process both donat kinetic energy to the new particle. At the center of a dense enough mass this will result in an initial singularity which then propagates outwards.
Title: Re: Lambert's Cosine Law
Post by: jeffreyH on 06/09/2015 13:07:19
There should be an uncertainty involved in this process as to whether a graviton will appear from the vacuum or return to the vacuum. Otherwise we have an infinite increase in the strength of gravity. This would be apparent at Lagrangian points which it obviously is not.